Title: Theory / Important / Math / Al gebra / Abstract Algebra - Dummit - Foote+14{ / 3 Module Theory
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On every note of this deck, unless a note says otherwise:
Chapter: 10

Name: A submodule
Definition: a subgroup of \(M\) closed under the action of \(R\)
Page: 337, 342
Chapter: 10
Section: 1
Variables: \((R,+,\times)\) a ring
\((M,+)\) a left \(R\)-module.
Index: , 1
Kind: Def, Prop
Definition2: a subset of \(M\) which is a module
Definition3: A nonempty subset \(N\) of \(M\) such that, \(x+ry\in N\) for all \(r\in R\) and \(x,y\in N\)

Name: A submodule
Definition: a subgroup of \(M\) closed under the action of \(R\)
Page: 337, 342
Chapter: 10
Section: 1
Variables: \((R,+,\times)\) a ring and
\((M,+)\) a left \(R\)-module.
Index: , 1
Kind: Def, Prop
Definition2: a subset of \(M\) which is a module
Definition3: A nonempty subset \(N\) of \(M\) such that, \(x+ry\in N\) for all \(r\in R\) and \(x,y\in N\)

Name: A left \(R\)-module
Definition: A group \((M,+)\) and a group action \(R\times M\to M\), distributive over both additions.
Page: 337, 338
Chapter: 10
Section: 1
Variables: \((R,+,\times)\) a ring.
Index: , (3)
Kind: Def, Ex
Example: \(R^n\), for \(n\in\mathbb N\)
Name2: Left module over \(R\)

Name: A unital left \(R\)-module
Definition: A left \(R\)-module, assuming \(R\) has a 1, in which case \(1m=m\).
Page: 337
Chapter: 10
Section: 1
Variables: \((R,+,\times)\) a ring.
Kind: Def

Smaller0: \(R\) is a field
Greater0: \(M\) is a \(R\)-vector space
Page: 337
Section: 1
Variables: \((R,+,\times)\) a ring
\((M,+)\) a left \(R\)-module.
Kind: Def
Smaller to greater: iff
Hide middle: yes

Smaller0: \(X\) is a submodule of \(R\)
Greater0: \(X\) is a left ideal of \(R\)
Page: 338
Section: 1
Variables: \((R,+,\times)\) a ring
\(X\subseteq R\).
Index: (1)
Kind: Ex
Smaller to greater: iff
Hide middle: yes

Q: A [...]-module is a [...]-module
A: \(R\); \(S\)
Page: 338
Chapter: 10
Section: 1
Variables: \((R,+,\times,1)\) a ring and
\((S,+,\times,1)\) a subring
Index: (4)
Kind: Ex

Name: A left \(R\)-module annihilated by \(I\)
Definition: A left \(R\)-module \(M\) such that \(IM=\{0\}\).
Page: 338
Chapter: 10
Section: 1
Variables: \((R,+,\times)\),
\(I\) an ideal of \(R\).
Index: (5)
Kind: Ex

Smaller0: \(I\) annihilates \(M\)
Greater0: \(M/I\) is an \(R/I\)-module
Page: 338
Section: 1
Variables: \((R,+,\times)\),
\(I\) an ideal of \(R\)
\(M\) a \(R\)-module
Index: (5)
Kind: Ex
Hide middle: yes

Smaller0: annihilates \(M\)
Greater0: an \(R/I\)-vector space
Page: 339
Section: 1
Variables: \((R,+,\times)\)
\(I\) an ideal of \(R\)
\(M\) a \(R\)-module
Index: (5)
Kind: Ex
Hide middle: yes
Prefix Greater: \(M/I\) is
Smaller1: is a maximal ideal of \(R\)
Greater1: a field
Connect Greater0: and \(R/I\) is
Prefix Smaller: \(I\)

Smaller0: \(N\) is a submodule
Greater0: \(N\) is a subgroup
Page: 339
Section: 1
Variables: \(M\) an Abelian group/\(\mathbb Z\)-module and
\(N\subseteq M\).
Kind: Ex
Smaller to greater: iff

Definition: \(a+\dots+a\) if \(a>n\) (\(n\) times), \(0\) if \(n=0\) and \(-a-\dots-a\) if \(a<n\) (\(-n\) times)
Page: 339
Chapter: 10
Section: 1
Variables: \(G\) an abelian group
\(n\in\mathbb Z\)
Notation: \(na\)
Kind: Ex

Smaller0: for all \(g\in G\), \(ng=0\)
Greater0: \(G\) is a \(\mathbb Z/n\mathbb Z\)-module
Page: 339
Section: 1
Variables: \(n\in\mathbb N^{+}\)
\(G\) a group
Assuming: \(G\) abelian
Smaller1: \(|G|=n\)
Connect Smaller0: or

Question: An \(F[x]\)-module is exactly characterized by
Answer: its set, which is a \(F\)-vector spaces, and the image of \(x\), which defines a linear transformation \(T\).
Page: 342
Chapter: 10
Section: 1
Variables: \(F\) a field
Kind: Ex

Smaller0: \(W\) is a subspace of \(V\)
Greater0: \(W\) is a submodule of \((V,T)\)
Page: 342
Section: 1
Variables: \(F\) a field,
\((V,T)\) a \(F[x]\)-module and
\(W\subseteq V\).
Kind: Ex
Smaller1: \(W\) is stable by \(T\)
Extra: Ne retrouve pas page 342 343

Name: An \(R\)-algebra
Definition: \((A,+,\times,0,1)\) is a ring with identity
Page: 342
Chapter: 10
Section: 1
Variables: \((R,+,\times,0, 1)\) a ring
Kind: Def
Definition2: \(f\) is a morphism
Example: \(R\)
Conjdef: A pair \(((A,+,\times,0,1),f:R\to A)\) such that
Definition3: \(f(R)\) is included in the centralizer of \(A\)
Example2: \(R[x,y,\dots,z]\)
Assuming: \(R\) commutative
Example3: \(RG\) for \(G\) a finite group
Example4: \(R^n\) for any \(n\in\mathbb N\)

Question: \(A\) is also
Answer: a left and right module defined by \(r\cdot a=a\cdot r=af(r)\).
Variables: \((R,+,\times,0, 1)\) a commutative ring
\((A,+_A,\times_A,0_A,1_A,f)\) an \(R\)-algebra
Index: 342
Kind: Def

Name: The torsion submodule
Definition: The set of torsion element.
Page: 344
Chapter: 10
Section: 1
Variables: \(R\) a ring,
\(M\) a \(R\)-module
Index: 8
Kind: Ex
Abbreviation: \(\text{Tor}(M)\)

Name: A \(R\)-algebra homo/iso-morphism
Definition: A ring homo/iso-morphism such that \(\phi(r\cdot a)=r\cdot \phi(a)\) for all \(r\in R\) and \(a\in A\).
Page: 343
Chapter: 10
Section: 1
Variables: \((R,+,\times,0, 1)\) a commutative ring,
\((A,+_A,\times_A,0_A,1_A)\), \((B,+_B,\times_B,0_B,1_B)\), two \(R\)- algebras and
\(\phi:A\to B\)
Kind: Def

Name: \(\mathbb Z\)-algebra
Page: 343
Chapter: 10
Section: 1
Index: (1)
Kind: Ex
Example: Any ring with identity

Smaller0: \(S\) is a subring of the center of \(R\)
Greater0: \((R,id)\) is an \(S\)-algebra
Page: 343
Section: 1
Variables: \(R\) a commutative ring
\(S\subseteq R\)
Index: (2)
Kind: Ex
Hide middle: yes
Smaller1: \(S\) contains the identity

Name: an \(R\)-module homomorphism
Definition: \(\phi(x+y)=\phi(x)+\phi(y)\) for \(x,y\in M\)
Page: 345
Chapter: 10
Section: 2
Variables: \(R\) a ring
\(M,N\) be \(R\)-modules
Index: (1)
Kind: Def
Definition2: \(\phi(rx)=r\phi(x)\) for \(x\in M\) and \(r\in R\)
Conjdef: Some \(\phi:M\to N\) with

Name: An \(R\)-module isomorphism
Definition: A bijective \(R\)-module homomorphism.
Page: 345
Chapter: 10
Section: 2
Variables: \(R\) a ring, \(M,N\) be \(R\)-modules and \(\phi:M\to N\).
Index: (2)
Kind: Def

Page: 345
Definition: There exists an isomorphism \(M\to N\).
Chapter: 10
Section: 2
Variables: \(R\) a ring, \(M,N\) be \(R\)-modules
Notation: \(M\cong N\)
Name: Isomorphic \(R\)-modules
Kind: Def
Index: (2)

Name: The kernel of \(\phi\)
Definition: The set of elements \(x\in M\) such that \(\phi(x)=0\).
Page: 345
Chapter: 10
Section: 2
Variables: \(R\) a ring
\(M,N\) be \(R\)-modules
\(\phi:M\to N\) an \(R\)-module homomorphism.
Index: (3)
Kind: Def
Typ: A \(R\)-submodule of \(M\)

Question: The image of \(\phi\) is
Answer: a \(R\)-submodule of \(N\)
Page: 345
Chapter: 10
Section: 2
Variables: \(R\) a ring,
\(M,N\) be \(R\)-modules and
\(\phi:M\to N\) an \(R\)-module homomorphism.
Index: (3)
Kind: Def

Definition: The set of \(R\)-homomorphism from \(M\) to \(N\)
Page: 345
Chapter: 10
Section: 2
Variables: \(R\) a ring, \(M,N\) be \(R\)-modules.
Notation: \(\text{Hom}_{R}(M,N)\)
Index: (4)
Kind: Def

Name: \(R\)-module homomorphism which is not a ring-homomorphism
Page: 346
Chapter: 10
Section: 2
Index: (1)
Kind: Ex
Example: \(\times 2\) on \(R=M=\mathbb Z\)

Name: Ring homomorphism which is not a \(R\)-module-homomorphism
Page: 346
Chapter: 10
Section: 2
Index: (1)
Kind: Ex
Example: \(f(x)\mapsto{}f(x^2)\) on \(R=M=F[x]\)

Smaller0: \(\phi\) is a \(\mathbb Z\)-module morphism
Greater0: \(\phi\) is a Abelian group morphism
Page: 346
Section: 2
Variables: \(M,N\) \(\mathbb Z\)-modules and
\(\phi:M\to N\)
Index: (4)
Kind: Ex
Smaller to greater: iff

Smaller0: \(I\) annihilates \(M\) and \(N\)
Greater0: \(\phi\) is a \(N/I\to M/I\) module
Page: 346
Section: 2
Variables: \(R\) a ring,
\(I\subseteq R\) an ideal,
\(M,N\) 2 \(R\)-modules and
\(\phi:N\to M\) a module homomorphism.
Index: (5)
Kind: Ex

Smaller0: \(G\) is abelian
Greater0: a \(\mathbb Z/p\mathbb Z\)-module homomorphism
Page: 346
Section: 2
Variables: \(G\) a group, \(p\) and \(\phi:G\to G\) an homomorphism.
Index: (5)
Kind: Ex
Hide middle: yes
Prefix Greater: \(\phi\) is
Smaller1: \(p\) prime
Smaller2: \(pG=\{0\}\)

Smaller0: \(G\) is abelian
Greater0: for \(G\) considered as a \(\mathbb Z/p\mathbb Z\)-module, \(GL(G)\)
Page: 346
Section: 2
Variables: \(G\) a group
Index: (5)
Kind: Ex
Assuming: prime \(p\)
Smaller1: \(pG=\{0\}\)
Greater1: \(\text{Aut}(G)\)
Connect Greater0: is equal to
Extra: group ?

Smaller0: \(R\) is commutative
Greater0: \(R\)-module
Page: 346
Section: 2
Variables: \(R\) a field,
\(M,N\) \(R\)-modules.
Index: 2 (2)
Kind: Prop
Hide middle: yes
Prefix Greater: \(\text{Hom}_R(M,N)\) is an

Definition: the element \(\text{Hom}_R(M,N)\), \(m\mapsto \phi(m)+r\psi(m)\).
Page: 346
Chapter: 10
Section: 2
Variables: \(R\) a ring
\(M,N\) some \(R\)-modules
\(r\in R\)
\(\phi,\psi:M\to N\).
Notation: \(\phi+r\psi\)
Index: 2 (2)
Kind: Prop

Name: \(R\)-module homomorphism
Page: 347
Section: 2
Chapter: 10
Index: 2 (3)
Variables: \(R\) a ring.
Kind: Prop
Under: composition

Page: 347
Chapter: 10
Section: 2
Variables: \(R\) a ring
\(M\) a \(R\)-module
Notation: \((\text{Hom}_R(M,M),+,\circ)\)
Index: 2 (4)
Kind: Prop
Typ: Ring with 1

Smaller0: \(R\) is commutative
Greater0: an \(R\)-algebra
Page: 347
Section: 2
Variables: \(R\) a ring
\(M\) a \(R\)-module.
Index: 2 (4)
Kind: Prop
Hide middle: yes
Prefix Greater: \((\text{Hom}_R(M,M),+,\circ)\) is

Name: The endomorphism ring of \(M\)
Definition: \(\text{Hom}_R(M,M)\).
Page: 347
Chapter: 10
Section: 2
Variables: \(R\) a ring
\(M\) a \(R\)-module
Notation: \(\text{End}_R(M)\)
Kind: Def

Name: A scalar transformations
Definition: A functions \(rI\in \text{End}_R(M)\), for \(r\in R\).
Page: 348
Chapter: 10
Section: 2
Variables: \(R\) a commutative ring
\(M\) a \(R\)-module.

Name: The subring of scalar transformations
Page: 348
Chapter: 10
Section: 2
Variables: \(R\) a commutative ring
\(M\) a \(R\)-module.
Typ: Subring of \(\text{End}_R(M)\)
Assuming: \(R\) is a field

Page: 348
Chapter: 10
Section: 2
Set: \(\{x+N\mid x\in M\}\)
Variables: \(R\) a ring,
\(M\) a \(R\)-module with submodule \(N\).
Notation: \(M/N\)
Kind: Prop
Index: 3
+: \(\left((x+N),(y+N)\right)\mapsto (x+y)+N\)
0: \(0+N\)
Typ: A \(R\)-module
×: \((r,(x+N))\mapsto rx+N\).

Definition: [$]\defFun{M}{M/N}{x}{x+N}[/$].
Page: 348
Chapter: 10
Section: 2
Variables: \(R\) a ring,
\(M\) a \(R\)-module
\(N\) a submodule.
Notation: \(\pi_N\)
Index: 3
Kind: Prop
Typ: A \(R\)-module homomorphism with kernel \(N\)

Name: \(A+B\)
Definition: the smallest submodule of \(M\) containing \(A\) and \(B\),
Page: 349
Chapter: 10
Section: 2
Variables: \(R\) a ring,
\(M\) a \(R\)-module
\(A,B\) submodules of \(M\)
Index: Def
Definition2: \(\{a+b\mid a\in A,b\in B\}\).

Name: \(\text{Ker}(\phi)\)
Page: 349
Chapter: 10
Section: 2
Variables: \(R\) a ring
\(M,N\) be \(R\)-modules
\(\phi:M\to R\) a \(R\)-module homomorphism
Index: 4 (1)
Kind: Theo
Typ: A submodule of \(M\)

Smaller0: \(M/\text{ker}(\phi)\)
Greater0: \(\phi(M)\)
Page: 349
Section: 2
Variables: \(R\) a ring,
\(M,N\) be \(R\)-modules
\(\phi:M\to R\) a \(R\)-module homomorphism
Index: 4 (1)
Kind: Theo
Smaller to greater: is isomorphic to
Name: First isomorphism theorem for modules

Smaller0: \((A+B)/B\)
Greater0: \(A/(A\cap B)\)
Page: 349
Section: 2
Variables: \(R\) a ring
\(M,N\) be \(R\)-modules with submodules \(A,B\)
Index: 4 (2)
Kind: Theo
Smaller to greater: is isomorphic to
Name: Second Isomorphism Theorem for Modules

Name: Third Isomorphism Theorem for Modules
Definition: \((M/A)/(B/A)\) is isomorphic to \((M/B)\).
Page: 349
Chapter: 10
Section: 2
Variables: \(R\) a ring,
\(M\) be \(R\)-modules with submodules \(A\subseteq B\).
Index: 4 (3)
Kind: Theo

Smaller0: the lattice of submodules of \(M/N\)
Greater0: the lattice of submodules of \(M\) which contains \(N\)
Page: 349
Section: 2
Variables: \(R\) a ring
\(M\) be \(R\)-modules
\(N\subseteq M\) a submodule.
Index: 4 (4)
Kind: Theo
Smaller to greater: is a lattice isomorphism with
Name: Fourth Isomorphism Theorem for Modules
Extra: lattice ?
Construction: the function sending a submodule \(A\) containing \(N\) to \(A/N\)
Name2: Lattice Isomorphism Theorem for Modules

Name: The sum of \(N_1,\dots,N_n\)
Definition: \(\{\sum a_i\mid a_i\in N_i\}\).
Page: 351
Chapter: 10
Section: 3
Variables: \(R\) a ring,
\(M\)a \(R\)-modules and
\(N_1,\dots,N_n\) submodules
Notation: \(N_1+\dots+N_n\)
Index: (1)
Kind: Def

Name: The submodule of \(M\) generated by \(A\)
Definition: \(\{\sum_{a\in A'} r_aa\mid A' \subseteq A\text{ finite}, (r_a)_{a\in A'}\in R\}\).
Page: 351
Chapter: 10
Section: 3
Variables: \(R\) a ring,
\(M\) a \(R\)-modules and
\(A\subseteq M\) any set
Notation: \(RA\)
Index: (2)
Kind: Def

Name: Generating set of \(N\)
Definition: A set \(A\subseteq M\) such that \(N=RA\).
Page: 351
Chapter: 10
Section: 3
Variables: \(R\) a ring,
\(M\)a \(R\)-modules
\(N\subseteq M\) a submodule.
Index: (2)
Kind: Def

Name: Finitely generated submodule
Definition: \(RA\) for some finite set \(A\subseteq M\).
Page: 351
Chapter: 10
Section: 3
Variables: \(R\) a ring
\(M\)a \(R\)-modules.
Index: (3)
Kind: Def

Name: A cyclic submodule of \(R\)
Definition: A submodule \(R\{a\}\) for some \(a\in M\).
Page: 351
Chapter: 10
Section: 3
Variables: \(R\) a ring,
Index: (4)
Kind: Def
Example: \(R=R1\)

Smaller0: \(\sum RA_i\)
Greater0: \(R(\bigcup A_i)\)
Page: 351
Section: 3
Variables: \(R\) a ring,
\(M\)a \(R\)-modules and
\(A_1,\dots,A_n\subseteq M\).
Smaller to greater: =

Name: A minimal set of generators for \(N\)
Definition: A set \(A\subseteq M\) such that \(N=RA\) and there is no \(B\) with \(|B|<|A|\) such that \(RB=RA\).
Page: 351
Chapter: 10
Section: 3
Variables: \(R\) a ring,
\(M\) a \(R\)-modules
and \(N\subseteq M\).

Q: [module notation] is [group notation].
A: \(\mathbb Z\{a\}\); \(\langle a\rangle\)
Page: 352
Chapter: 10
Section: 3
Variables: \(M\) a \(\mathbb Z\)-module and \(a\in M\).
Index: (1)
Kind: Ex

Q: [module notation] is [group notation].
A: \(\mathbb ZA\); \(\langle A\rangle\)
Page: 352
Chapter: 10
Section: 3
Variables: \(M\) a \(\mathbb Z\)-module
\(A\subseteq M\).
Index: (1)
Kind: Ex

Smaller0: \(X\) is a principal ideal of \(R\)
Greater0: \(X\) is a cyclic \(R\)-submodule of \(R\)
Page: 352
Section: 3
Variables: \(R\) a ring with unit
\(X\subseteq R\).
Index: (2)
Kind: Ex
Smaller to greater: iff

Smaller0: \(X\) is a finitely generated ideal of \(R\)
Greater0: \(X\) is a finitely generated \(R\)-submodule of \(R\)
Page: 352
Section: 3
Variables: \(R\) a ring with unit
\(X\subseteq R\).
Index: (2)
Kind: Ex
Smaller to greater: iff

Name: A Principal Ideal Domain
Definition: An integral domain \(R\) with identity in which every \(R\)-submodule of \(R\) is cyclic.
Page: 352
Chapter: 10
Section: 3
Index: (2)
Kind: Ex
Extra: i.e. ideal are principal

Name: Non-finitely-generated \(R\)-submodule \(N\) of a finitely generated \(R\)-module \(M\)
Page: 352
Chapter: 10
Section: 3
Variables: \(R\) a ring,
\(M\) a module with submodule \(N\)
Index: (2)
Kind: Ex
Example: \(R=\mathbb Z[\left(x_i\right)_{i\in\mathbb N}]\), \(M=R\) and \(N=R\left\{x_i\mid x\in \mathbb N\right\}\)

Name: The natural map from \(R\) to \(\text{End}(M)\)
Definition: \(r\mapsto rI\)
Page: 348
Chapter: 10
Section: 2
Variables: \(R\) a ring
Assuming: \(R\) is commutative

Smaller0: \(N\) have a finite set of generators
Greater0: \(N\) has a minimal set of generators
Page: 351
Section: 3
Variables: \(R\) a ring,
\(M\) a \(R\)-modules
\(N\subseteq M\).
Smaller to greater: iff
Assuming: \(R\) is commutative

Name: A torsion element
Definition: An element \(m\in M\) such that \(rm=0\) for some nonzero element \(r\in R\).
Page: 344
Chapter: 10
Section: 1
Variables: \(R\) a ring,
\(M\) a \(R\)-module
Index: 8
Kind: Ex

Name: The annihilator of \(N\)
Definition: \(\{r\in R\mid rn=0\text{ for all }n\in N\}\).
Page: 344
Chapter: 10
Section: 1
Variables: \(R\) a ring,
\(M\) a \(R\)-module,
\(N\subseteq M\) a submodule
Index: 9
Kind: Ex

Name: The annihilator of \(I\)
Definition: \(\{m\in M\mid am=0\text{ for all }a\in I\}\).
Page: 344
Chapter: 10
Section: 1
Variables: \(R\) a ring,
\(M\) a \(R\)-module,
\(I\subseteq R\) a right submodule
Index: 9
Kind: Ex

Name: Affine \(n\)-space over \(F\)
Definition: The set of \(n\)-tuples of elements of \(F\).
Page: 338
Chapter: 10
Section: 1
Variables: \(F\) a field
Notation: \(F^n\)
Kind: (2)

Question: Exemple of a generating set for \(R^n\)
Answer: The set of unit vectors.
Page: 352
Chapter: 10
Section: 3
Variables: \(n\in\mathbb N\),
\(R\) a ring with \(1\),
Kind: Ex

Smaller0: cyclic
Greater0: \(V=\{p(x)v\mid p(x)\in F[x]\}\) for \(v\) a generator
Page: 352
Section: 3
Variables: \(F\) a field,
\(x\) an indeterminate,
\(V\) a \(F\)-vector space
\(T:V\to V\) linear,
Index: (4)
Kind: Ex
Smaller to greater: iff
Prefix Smaller: the \(F[x]\)-module \((V,T)\) is
Greatest: \(\{p^i(T)\mid i\in\mathbb N\}\) spans \(V\)

Name: A linear function \(T\) such that \(V\) is \(F[x]\)-cyclic
Page: 352
Chapter: 10
Section: 3
Variables: \(F\) a field,
\(x\) an indeterminate,
\(n\in\mathbb N\)
\(V\) a \(F\)-vector space with standard basis \(e_1,\dots,e_n\)
Index: (4)
Kind: Ex
Example: The shift operation

Name: The shift operator
Definition: [$]\defFun{V}{V}{(a_1,\dots, a_n)}{(a_2,\dots,a_n,0)}[/$].
Page: 340
Chapter: 10
Section: 1
Variables: \(F\) a field,
\(n\in\mathbb N\)
\(V\) a \(F\)-vector space with standard basis \(e_1,\dots,e_n\)

Page: 357
Chapter: 10
Section: 3
Name: The direct sum of the \(M_j\)
Set: subset of \(\prod_{j}M_j\) with a finite support
Variables: \(R\) a ring,
Let \(J\) an index set
\((M_j,+_j,\cdot_j)_{j\in J}\) a collection of \(R\)-modules
Notation: \(\bigoplus_{j\in J} M_j\)
Kind: Exercice
Index: 20
+: Component wise
Typ: \(R\)-module
Name2: The external direct sum of the \(M_j\)

Page: 357
Chapter: 10
Section: 3
Name: The direct product of the \(M_j\)
Set: \(\prod_{j\in J} M_j\)
Variables: \(R\) a ring,
\(J\) an index set
\((M_j,+_j,\cdot_j)_{j\in J}\) a collection of \(R\)-modules
+: The direct product as a group
0: tuple of 0
Typ: \(R\)-module
×: action of \(R\) component-wise.
Ring-base: \(R\)

Smaller0: \(\pi\) is an isomorphism
Greater0: \(N_i\cap \bigoplus_{j\in J\setminus \{i\} } N_j=\{0\}\) for each \(i\in I\)
Page: 353
Section: 3
Variables: \(M\) a module,
\(J\) an index set,
\((N_j)_{j\in J}\) submodules
\(\pi\) defined on sequences \((a_j)_{j\in J}\) with finite support, such that \(\pi((a_j)_{j\in J})=\sum_{j\in J} a_j\)
Index: 5
Kind: Prop
Smaller to greater: iff
Extra: \(J\) finite in the proposition
Greatest: every \(x\in \bigoplus_{j\in J} N_j\) can be written uniquely in the form \(\sum_{j\in J} a_j\)

Name: \(M\) is the internal direct sum of the \(N_j\)'s
Definition: Each element of \(M\) can be written uniquely as a sum of elements of \((N_j)_{j\in J}\).
Page: 354
Chapter: 10
Section: 3
Variables: \(M\) a module,
\(J\) an index set,
\((N_j)_{j\in J}\) submodules
Index: 5
Kind: Prop
Extra: \(J\) finite in the proposition

Name: A free \(R\)-module on the subset \(A\)
Definition: A \(R\)-module \(F\supseteq A\) such that for every nonzero element \(x\in F\), there exists unique nonzero elements
\(r_1,\dots,r_n\in R\) and \(a_1,\dots,a_n\in A\) such that \(x=\sum_{i=1}^n r_ia_i\).
Page: 354
Chapter: 10
Section: 3
Variables: \(R\) a ring,
\(A\) a set

Name: A basis for \(F\)
Page: 354
Chapter: 10
Section: 3
Variables: \(R\) a ring,
\(A\) a set,
\(F\) a free \(R\)-module on the subset \(A\)
Kind: Def
Example: \(A\).
Name2: A set of free generators for \(F\)

Name: The rank of \(F\)
Definition: The cardinality of \(A\).
Page: 354
Chapter: 10
Section: 3
Variables: \(R\) a ring,
\(A\) a set,
\(F\) a free \(R\)-module on the subset \(A\)
Kind: Def
Assuming: \(R\) is commutative

Name: The free \(R\)-module on the set \(A\)
Definition: such that if \(M'\) is any \(R\)-module and \(\phi:A\to M'\) is any map of sets there is a unique \(R\)-module homomorphism \(\Phi:M\to M'\) such that \(\Phi(a)=\phi(a)\) for all \(a\in A\).
Page: 354
Chapter: 10
Section: 3
Variables: \(R\) a ring
\(A\) a set
Notation: \(F(A)\)
Index: 6
Kind: Theo
Definition2: isomorphic to \(\bigoplus_{a \in A} Ra\simeq R^{|A|}\) (assuming \(A\) finite)
Conjdef: An \(R\)-module \(M\)

Question: The set of isomorphism between \(F(A)\) and \(F'(A)\)
Answer: A singleton, which is the identity on \(A\).
Page: 354
Chapter: 10
Section: 3
Variables: \(R\) a ring,
\(A\) a set,
\(F(A)\), \(F'(A)\) two free \(R\)-modules on the set \(A\)
Index: 7(1)
Kind: Cor

Name: The free abelian group on \(A\)
Definition: The free \(\mathbb Z\)-module on the set \(A\).
Page: 354
Chapter: 10
Section: 3
Variables: \(A\) a set,

Name: The free abelian group of rank \(n\)
Definition: \(\oplus_{i=1}^n\mathbb Z\).
Page: 355
Chapter: 10
Section: 3
Variables: \(n\in\mathbb N\)

Name: A torsion module
Definition: A \(R\)-module \(M\) such that each elements are torsion elements.
Page: 355
Chapter: 10
Section: 3
Variables: \(R\) a ring
Index: 4
Kind: Exercice

Name: An irreducible module
Definition: A \(R\)-module \(M\ne 0\) such that, 0 is the only proper submodule of \(M\).
Page: 355
Chapter: 10
Section: 3
Variables: \(R\) a ring
Index: 9
Kind: Exercice

Name: A central idempotent
Definition: An idempotent in the centralizer of \(R\).
Page: 357
Chapter: 10
Section: 3
Variables: \(R\) a ring
Index: 15
Kind: Exercice

Name: The \(R\)-module obtained from \(N\) by restriction of scalars from \(S\) to \(R\)
Definition: \(\left(N,+_N,(r,n)\mapsto f(r)\circ_{(S,N)}n\right)\)
Page: 359
Chapter: 10
Section: 4
Variables: \(R,S\) rings with unity,
\(\left(N,+_N,\circ_{(S,N)}\right)\) a left \(S\)-module,
\(f:R\to S\) a ring homomorphism
Assuming: \(R\) commutative

Page: 360
Section: 4
Notation: \(S\otimes_RN\)
Variables: \(R\subseteq S\) rings with unity,
\(\left(N,+_N,\circ_{(R,N)}\right)\) a left \(R\)-module,
Chapter: 10
Name: The tensor product of \(S\) and \(N\) over \(R\)
Definition: The quotient of \(\mathbb Z(S\times N)\) by the subgroup generated by\((s_1+s_2,n)-(s_1,n)-(s_2,n)\), \((s,n_1+n_2)-(s,n_1)-(s,n_2)\) and \((sr,n)-(s,rn)\) under the action \(\left(s,\left(\sum s_i\otimes n_i\right)\right)\mapsto \sum (ss_i)\otimes n_i\)
Codomain: An \(S\)-left module

Name: The tensor of \(s\) and \(n\)
Definition: The coset of \((s,n)\) in the tensor product of \(S\) and \(N\) over \(R\)
Page: 360
Chapter: 10
Section: 4
Variables: \(R\subseteq S\) rings with unity,
\(N\) a left \(R\)-module,
\(s\in S\),
\(n\in N\)
Notation: \(s\otimes n\)

Name: The natural map of tensor product of module
Definition: [$]\defFun{N}{S\otimes_RN}{n}{1_S\otimes n}[/$]
Page: 361
Chapter: 10
Section: 4
Variables: \(R\subseteq S\) rings,
\(N\) a left \(R\)-module
Index: last paragraph
Denoted by: \(\iota\)

Name: Universal property of tensor product of module
Definition: An universal pair \((L,\phi)\) where \(L\) is a left \(S\)-module and \(\phi:N\to L\) is an \(R\)-module homomorphism.
Page: 362
Chapter: 10
Section: 4
Variables: \(R\subseteq S\) rings,
\(N\) a left \(R\)-module
Index: 8
Kind: Theo

Question: The largest quotient of \(N\) that can be embedded in any \(S\)-module is
Answer: \(N/\text{ker }\iota\) with \(\iota:N\to S\otimes_RN\)
Page: 362
Chapter: 10
Section: 4
Variables: \(R\subseteq S\) rings,
\(N\) a left \(R\)-module
Index: 9
Kind: Cor

Smaller0: \(N\) can be embedded as an \(R\)-submodule of some left \(S\)-module
Greater0: \(\iota:N\to S\otimes_RN\) is injective
Page: 362
Section: 4
Variables: \(R\subseteq S\) rings,
\(N\) a left \(R\)-module
Index: 9
Kind: Cor
Smaller to greater: iff

Question: \(R\otimes_RN\) is
Answer: isomorphic to \(N\)
Page: 363
Chapter: 10
Section: 4
Variables: \(R\) a ring,
\(N\) a left \(R\)-module
Index: 1
Kind: Example

Question: \(\mathbb Q\otimes_{\mathbb Z}A\)
Answer: equals 0
Page: 363
Chapter: 10
Section: 4
Variables: \(A\) a finite abelian group
Index: 2
Kind: Example

Question: \(S\otimes_RR^n\)
Answer: equals \(S^n\)
Page: 363
Chapter: 10
Section: 4
Variables: \(n\in\mathbb N\)
\(R\subseteq S\) rings
Index: 3
Kind: Example

Name: The tensor product of \(M_l\) and \(N_r\) over \(R\)
Definition: \((m^r_1+m^r_2,m^l)-(m^r_1,m^l)-(m^r_2,m^l)\) and \((m^r,m^l_1+m^l_2)-(m^r,m^l_1)-(m^r,m^l_2)\)
Page: 364
Chapter: 10
Section: 4
Variables: \(R\) a ring
\(M_l\) a left \(R\)-module
\(M_r\) a right-\(R\) module
Notation: \(M_r\otimes_RM_l\)
Kind: Example
Definition2: \((m^rr,m^l)-(m^r,rm^l)\)
Conjdef: The quotient of the free \(\mathbb Z\)-module on the set \(M_r\times M_l\) by the subgroup generated by all elements of the form

Name: The tensors
Definition: The elements of \(M_r\otimes_RM_l\)
Page: 364
Chapter: 10
Section: 4
Variables: \(R\) a ring
Let \(M_l\) a left \(R\)-module
\(M_r\) a right-\(R\) module
Kind: Example

Name: A simple tensor
Definition: An element of the form \(m_r\otimes m_l\) with \(m_i\in M_i\)
Page: 364
Chapter: 10
Section: 4
Variables: \(R\) a ring
Let \(M_l\) a left \(R\)-module
\(M_r\) a right-\(R\) module
Kind: Example

Question: A tensor can be written as
Answer: a sum of simple tensors
Page: 364
Chapter: 10
Section: 4
Variables: \(R\) a ring
Let \(M_l\) a left \(R\)-module
\(M_r\) a right-\(R\) module

Question: The number of way to write tensor as sums of simple tensors is
Answer: at least one
Page: 364
Chapter: 10
Section: 4
Variables: \(R\) a ring
\(M_l\) a left \(R\)-module
\(M_r\) a right-\(R\) module

Question: A function \(m_r\otimes m_l\mapsto f(m_r,m_l)\) is
Answer: generally not well defined, because \(m_r\otimes m_l=m'_r\otimes m'_l\) does not imply \(m_r=m'_r\), \(m_l=m_l'\).
Page: 364
Chapter: 10
Section: 4
Variables: \(R\) a ring
\(M_l\) a left \(R\)-module
\(M_r\) a right-\(R\) module

Name: A \(R\)-balanced map
Definition: A map \(\phi:M_r\times M_l\to L\), additive in both component, such that \(\phi(m_r,rm_l)=\phi(m_nr,m_l)\)
Page: 365
Chapter: 10
Section: 4
Variables: \(R\) a ring
\(M_l\) a left \(R\)-module
\(M_r\) a right-\(R\) module
\((L,+)\) an abelian group
Kind: Def
Name2: A middle linear map with respect to \(R\)

Question: \(\Phi\circ\iota\) (of type \(M\times N\to L\)) is
Answer: an \(R\)-balanced map
Page: 365
Chapter: 10
Section: 4
Variables: \(R\) a ring
\(M_l\) a left \(R\)-module
\(M_r\) a right-\(R\) module
\((L,+)\) an abelian group
\(\iota:M\times N\to M\otimes_RN\)
\(\Phi:M_r\otimes _RM_l\to L\) a group homomorphism

Name: Universal property of tensor product
Definition: \(M_r\otimes _RM_l\) an Abelian group and
Page: 365, 366
Chapter: 10
Section: 4
Variables: \(R\) a ring
Let \(M_l\) a left \(R\)-module
\(M_r\) a right-\(R\) module
Index: 10, 11
Kind: Theo, Cor
Definition2: \(\iota:M_r\times M_l\to M_r\otimes _RM_l\) an \(R\)-balanced map
Conjdef: A pair with
Definition3: such that for each Abelian group \((L,+)\) and each \(R\)-balanced map \(\phi:M_r\times M_l\to L\), there exists a unique \(\Phi:M_r\times M_l\to L\) such that \(\phi=\Phi\circ \iota\)

Smaller0: \(\iota'\) generates \(D\) as an Abelian group
Greater0: there is an isomorphism \(f:M_r\otimes _RM_l\to D\) such that \(\iota'=f\circ\iota\) where \(\iota:M_r\times M_l\to{M_r\otimes_RM_l}\) is the natural map of tensor product of module
Page: 366
Section: 4
Variables: \(R\) a ring with \(1\)
\(M_l\) a left \(R\)-module
\(M_r\) a right-\(R\) module
\(D\) an Abelian group
\(\iota':M\times N\to D\)
Index: 11
Kind: Cor
Assuming: \(\iota'\) is \(R\)-balanced,
Smaller1: every \(R\)-balanced map defined on \(M_r\times M_l\) factors through \(\iota'\)

Name: An \((S,R)\)-bimodule
Definition: is a left \(S\)-module, a right \(R\)-module and
Page: 366, 367
Chapter: 10
Section: 4
Variables: \(R,S\) rings with \(1\)
Index: , 1, 2, 3
Kind: Def,Ex
Definition2: such that \(s(mr)=(sm)r\).
Conjdef: An Abelian group \(M\) which

Name: The standard \(R\)-module structure on \(M\)
Definition: An \((R,R)\)-bimodule with:
elements \(M\)
\((r,m)\mapsto r\cdot_{RM}m\)
\((m,r)\mapsto r\cdot_{RM}m\)
Page: 367
Chapter: 10
Section: 4
Variables: \((R,+_R,\times_R)\) a ring
\((M,+_M,\cdot_{RM})\) a left \(R\)-module
Assuming: \(R\) commutative

Definition: \((m\otimes n)\mapsto sm\otimes n\)
Page: 367
Chapter: 10
Section: 4
Variables: \(R\) a ring
\(N\) a left-module
\(M\) an \((S,R)\)-bimodule
\(s\in S\)
Notation: \(\lambda s\)

Question: The standard \(R\)-module structure on \(M\) gives \(M\otimes_RN\)
Answer: the structure of an left \(R\)-module
Page: 367
Chapter: 10
Section: 4
Context: \(R\) commutative

Name: An \(R\)-bilinear map
Definition: \(\phi(r_1m_1+r_2m_2,n)=r_1\phi(m_1,n)+r_2\phi_2(m_2,n)\)
Page: 368
Chapter: 10
Section: 4
Variables: \(R\) a ring,
\(M,N,L\) left \(R\)-modules
Kind: Def
Definition2: \(\phi(m,r_1n_1+r_2n_2)=r_1\phi(m,n_1)+r_2\phi_2(m,n_2)\)
Conjdef: A map \(\phi:M\times N\to L\)
Assuming: \(R\) commutative

Smaller0: \(r(m\otimes n)\)
Greater0: \((rm)\otimes n\)
Page: 368
Section: 4
Variables: \(R\) a ring,
\(M,N\) left \(R\)-modules
\(M\otimes_RN\) the left-module with standard structure
Index: 12
Kind: Cor
Smaller to greater: =
Assuming: \(R\) commutative
Smallest: \((mr)\otimes n\)
Greatest: \(m\otimes (rn)\)

Question: [$]\defFun{M\times N}{M\otimes_RN}{(m,n)}{m\otimes n}[/$] is
Answer: \(R\)-bilinear
Page: 368
Chapter: 10
Section: 4
Variables: \(R\) a ring,
\(M,N\) left \(R\)-modules
\(M\otimes_RN\) the left-module with standard structure
Index: 12
Kind: Cor

Smaller0: the set of \(R\)-bilinear maps \(M\times N\to L\)
Greater0: the set of \(R\)-module homomorphisms maps \(M\otimes_R N\to L\)
Page: 368
Section: 4
Variables: \(R\) a ring,
\(M,N,L\) left \(R\)-modules
\(M\otimes_RN\) the left-module with standard structure
Index: 12
Kind: Cor
Smaller to greater: is in bijection with

Question: \(m\otimes 0\)
Answer: equals \(0\)
Page: 368
Chapter: 10
Section: 4
Variables: \(R\) a ring,
\(M,N,L\) left \(R\)-modules
\(M\otimes_RN\) the left-module with standard structure
\(m\in M\),
\(n\in N\)
Index: 1
Kind: Ex

Question: \(0\otimes n\)
Answer: equals \(0\)
Page: 368
Chapter: 10
Section: 4
Variables: \(R\) a ring,
\(M,N,L\) left \(R\)-modules
\(M\otimes_RN\) the left-module with standard structure
\(m\in M\),
\(n\in N\)
Index: 1
Kind: Ex

Question: \(\mathbb Z/m\mathbb Z\otimes_{\mathbb Z}\mathbb Z/n\mathbb Z\)
Answer: is isomorphic to \(\mathbb Z/\gcd(m,n)\mathbb Z\)
Page: 369
Chapter: 10
Section: 4
Index: 2
Kind: Ex
Construction: \(a\otimes b=a\otimes (b\cdot 1)=(ab)\otimes 1=ab(1\otimes 1)\) hence \((1\otimes 1)\) is a generator.
\(m(1\otimes 1)=m\otimes 1=0\otimes 1=0\) hence the order divides \(n\) and \(m\).
[$]\defFun{\mathbb Z/m\mathbb Z\otimes_{\mathbb Z}\mathbb Z/n\mathbb Z}{\mathbb Z/\gcd(m,n)\mathbb Z}{(m,n)}{m\cdot n\mod \gcd (m,n)}[/$] is well defined, hence \(1\otimes 1\) is of order \(d\)

Question: The set of balanced/bilinear maps from \(\mathbb Z/2\mathbb Z\otimes_{\mathbb Z}\mathbb Z/3\mathbb Z\) to \(G\) is
Answer: the singleton {null map}
Page: 368
Chapter: 10
Section: 4
Variables: \(G\) an Abelian group
Index: 2
Kind: Ex

Question: \(\mathbb Q/\mathbb Z\otimes_{\mathbb Z}\mathbb Q/\mathbb Z\)
Answer: is isomorphic to \(0\)
Page: 369
Chapter: 10
Section: 4
Kind: Ex

Question: \(\mathbb{Q\otimes_Q Q}\) is
Answer: a one dimensional \(\mathbb Q\)-vector space
Page: 368
Chapter: 10
Section: 4
Index: 5
Kind: Ex

Question: \(\mathbb{Q\otimes_Z Q}\) is
Answer: a one dimensional \(\mathbb Q\)-vector space
Page: 368
Chapter: 10
Section: 4
Index: 5
Kind: Ex

Question: \(\mathbb{C\otimes_C C}\) is
Answer: a one dimensional \(\mathbb C\)-vector space
Page: 368
Chapter: 10
Section: 4
Index: 5
Kind: Ex

Question: \(\mathbb{C\otimes_R C}\) is
Answer: a two dimensional \(\mathbb C\)-vector space
Page: 368
Chapter: 10
Section: 4
Index: 5
Kind: Ex
Extra: No construction given in the proof

Name: The change of base of \(N\) from \(R\) to \(S\)
Definition: The \(S\)-module \(S\otimes_{R}N\), where \(S\) is the right \(R\)-module with rule \(s\cdot r=sf(r)\)
Page: 368
Chapter: 10
Section: 4
Variables: \(R,S\) rings
\(f:R\to S\) morphism preserving identity
\(N\) a left \(R\)-module
Index: 6
Kind: Ex
Name2: The general extension of scalar

Question: \(S\otimes_RR\)
Answer: is isomorphic to \(S\) as left \(S\)-module
Page: 368
Chapter: 10
Section: 4
Variables: \(R,S\) rings
\(f:R\to S\) morphism preserving identity
Index: 7
Kind: Ex

Definition: the set of finite sum \(\sum a_in_i\) with \(a_i\in I\) and \(n_i\in N\)
Page: 368
Chapter: 10
Section: 4
Variables: \(R\) ring
\(I\subseteq R\) two-sided ideal
\(N\) a left \(R\)-module
Notation: \(IN\)
Index: 7
Kind: Ex
Typ: left \(R\)-submodule of \(N\)

Name: The tensor product of \(\phi_l\) and \(\phi_r\)
Definition: [$]\defFun{M_r\otimes_RM_l}{M'_r\otimes M'_l}{m_r\otimes m_l}{\phi_r(m_r)\otimes\phi_l(m_l)}[/$]
Page: 368
Chapter: 10
Section: 4
Variables: \(R\) ring
\(M_r,M'_r\) right \(R\)-modules
\(M_l,M'_l\) left \(R\)-modules
\(\phi_r:M_r\to M_r'\), \(\phi_l:M_l\to M_l'\),
Notation: \(\phi_r\otimes\phi_l\)
Index: 13
Kind: Theo

Question: \([\mathbb F_{p^n}:\mathbb F_p]\) equals
Answer: \(n\)
Page: 549
Chapter: 13
Section: 5
Variables: \(p\) prime
\(n\in\mathbb N\)
Kind: Ex

Question: How to consider \(S\) as a right \(R\)-module
Answer: takes \(s\cdot r=s\times_Sf(r)\)
Page: 366
Chapter: 10
Section: 4
Variables: \((R,+_R,\times_S),(S,+_S,\times_S)\) rings with \(1\)
\(f:R\to S\) ring homomorphism
Index: 1
Kind: Ex

Name: \(S\) as a right \(R\)-module
Definition: the right \(R\)-module with
\((s,r)\mapsto s\times_Sf(r)\)
Page: 366
Chapter: 10
Section: 4
Variables: \((R,+_R,\times_R),(S,+_S,\times_S)\) rings with \(1\)
\(f:R\to S\) ring homomorphism
Index: 1
Kind: Ex

Question: \(R/I\) can be considered as a
Answer: \((R/I,R)\)-bimodule
Page: 366
Chapter: 10
Section: 4
Variables: \((R,+_R,\times_S)\) rings with \(1\)
\(I\subseteq R\) an ideal
Index: 1
Kind: Ex

Question: Example of \((R/I,R)\)-bimodule ?
Answer: \(R/I\)
Page: 366
Chapter: 10
Section: 4
Variables: \((R,+_R,\times_R)\) ring
\(I\subseteq R\) an ideal
Index: 2
Kind: Ex

Question: Example of \((R,R)\)-bimodule ?
Answer: \((m,r)\mapsto r\cdot_{RM}m\) and
\((r,m)\mapsto r\cdot_{RM}m\)
Page: 366
Chapter: 10
Section: 4
Variables: \((R,+_R,\times_R)\)
\((M,+_M,\cdot_{RM})\) a left-module
Index: 1
Kind: Ex
Assuming: \(R\) commutative

Name: \(M\) as a \((R,R)\)-bimodule
Definition: The bimodule with elements \(M\) and actions
\((m,r)\mapsto r\cdot_{RM}m\) and
\((r,m)\mapsto r\cdot_{RM}m\)
Page: 366
Chapter: 10
Section: 4
Variables: \((R,+_R,\times_R)\)
\((M,+_M,\cdot_{RM})\) a left-module
Index: 1
Kind: Ex
Assuming: \(R\) commutative

Name: \(M\) as an \((S,R)\)-bimodule
Definition: The bimodule with elements \(M\) and actions
\((s,m)\mapsto s\cdot_{SM}m\) and
\((m,r)\mapsto r\cdot_{SM}m\)
Page: 366
Chapter: 10
Section: 4
Variables: \((S,+_S,\times_S)\) rings
\((M,+_M,\cdot_{MS})\) a left-\(S\)-module
\(R\subseteq S\)
Index: 1
Kind: Ex
Assuming: \(R\subseteq\text{center}(S)\)

Name: \(R/I\) as a \((R/I,R)\)-bimodule
Definition: the \((R/I,R)\)-bimodule with elements \(R/I\)
with \((r+I,r'+I)\mapsto (r+r'+I)\) and
\((r+I,r')\mapsto (r+r'+I)\)
Page: 366
Chapter: 10
Section: 4
Variables: \((R,+_R,\times_S)\) rings with \(1\)
\(I\subseteq R\) an ideal
Index: 2
Kind: Ex

Page: 366
Chapter: 10
Section: 4
Name: \(M\otimes_RN\) as a left \(S\)-module
Set: the left \(S\)-module with
elements \(M\otimes_RN\)
and \((s,\sum m_i\otimes n_i)\mapsto\sum(sm_i)\otimes n_i\)
Variables: \((R,+_R,\times_R),(S,+_S,\times_S)\) rings
\((N,+_N,\cdot_{RN})\) left \(R\)-modules
\((M,+_M,\cdot_{RM}),\)
Kind: Eq
Index: 8

Smaller0: \(A\) divisible
Greater0: is isomorphic to 0
Page: 369
Section: 4
Variables: \(A,B\) groups
Kind: Ex
Prefix Greater: \(A\otimes_{\mathbb Z}B\)
Assuming: \(A,B\) Abelian
Smaller1: \(B\) torsion
Extra: Take \(A,B=\mathbb {Q/Z}\)
\((\frac ab\mod\mathbb Z)\otimes(\frac cd\mod\mathbb Z)\) equals
\(d(\frac a{bd}\mod\mathbb Z)\otimes(\frac cd\mod\mathbb Z)\) equals
\((\frac a{bd}\mod\mathbb Z)\otimes(d\frac cd\mod\mathbb Z)\) equals
\((\frac a{bd}\mod\mathbb Z)\otimes(0\mod\mathbb Z)\) equals 0
Construction: take \(a\in A\), \(b\in B\), written additively. \(n\) such that \(nb=0\).
\(a\otimes b=n(\frac an)\otimes b=\frac an\otimes nb=\frac an\otimes0=0\)

Name: The Abelian group of type \((n_1,\dots,n_s)\)
Definition: \(\prod\mathbb Z_{n_i}\)
Page: 163
Chapter: 5
Section: 2
Variables: \((n_1,\dots,n_s)\in\mathbb N\)
Kind: Ex
Assuming: nothing

Question: The elementary divisors of the Abelian group of type \((n_1,\dots,n_s)\) are
Answer: the elementary divisors of the \(\mathbb Z_{n_i}\)
Page: 163
Chapter: 5
Section: 2
Variables: \((n_1,\dots,n_s)\in\mathbb N\)
Kind: Ex

Page: 164
Implementation: For each prime \(p\), take \(a^i_p\) the list of \(e_i\) which is a power of \(p\), in non-decreasing order
Let \(t\) the length of the longest list.
Return the list of \((\prod_pa^i_p)_{i=1}^t\), with \(a_p^i\) being 1 if it is not defined
Section: 2
Chapter: 5
Effect: The invariant factors of \(G\)
Variables: \(G\) a group
\(e_1,\dots, e_m\) the elementary divisors of \(G\)
Kind: Ex

Name: The rank of \(G\)
Definition: the length of the type of \(G\)
Page: 165
Chapter: 5
Section: 2
Variables: \(G\) a group
Kind: Ex
Assuming: \(G\) finite Abelian

Name: The type of \(G\)
Definition: The list of invariant factor of \(G\), in non-decreasing order
Page: 165
Chapter: 5
Section: 2
Variables: \(G\) a group
Kind: Ex
Assuming: \(G\) finite Abelian

Name: The exponent of \(G\)
Definition: The least \(n\) such that \(x^n=1\) for each \(x\in G\)
Page: 165
Chapter: 5
Section: 2
Variables: \(G\) a group
Kind: Ex