Glossary of module theory

Module theory is the branch of mathematics in which modules are studied. This is a glossary of some terms of the subject.

Basic definition

left R-module

A left module

M

over the ring

R

is an abelian group

(M, +)

with an operation

R \times M \to M

(called scalar multipliction) satisfies the following condition:

\forall r,s \in R, m,n \in M

,

$r (m + n) = rm + rn$
$r (s m) = (r s) m$
$1_R \, m = m$

right R-module

A right module

M

over the ring

R

is an abelian group

(M, +)

with an operation

M \times R \to M

satisfies the following condition:

\forall r,s \in R, m,n \in M

,

$(m + n) r = m r + n r$
$(m s) r = r (s m)$
$m 1_R = m$

Or it can be defined as the left module

M

over

R^\textrm{op}

(the opposite ring of

R

).

bimodule: If an abelian group $M$ is both a left $S$ -module and right $R$ -module, it can be made to a $(S,R)$ -bimodule if $s(mr) = (sm)r \, \forall s \in S, r \in R, m \in M$ .

submodule: Given $M$ is a left $R$ -module, a subgroup $N$ of $M$ is a submodule if $RN \subseteq N$ .

homomorphism of $R$ -modules: For two left $R$ -modules $M_{1},M_{2}$ , a group homomorphism $\phi: M_1 \to M_2$ is called homomorphism of $R$ -modules if $r \phi(m) = \phi (r m) \, \forall r \in R, m \in M_1$ .

quotient module: Given a left $R$ -modules $M$ , a submodule $N$ , $M/N$ can be made to a left $R$ -module by $r(m+N) = rm + N \, \forall r \in R, m \in M$ . It is also called a factor module.

annihilator: The annihilator of a left $R$ -module $M$ is the set $\textrm{Ann}(M) := \{ r \in R | rm = 0 \, \forall m \in M \}$ . It is a (left) ideal of $R$ .; The annihilator of an element $m\in M$ is the set $\textrm{Ann}(m) := \{ r \in R | rm = 0 \}$ .

Types of modules

finitely generated module: A module $M$ is finitely generated if there exist finitely many elements $x_{1},...,x_{n}$ in $M$ such that every element of $M$ is a finite linear combination of those elements with coefficients from the scalar ring $R$ .

cyclic module: A module is called a cyclic module if it is generated by one element.

free module

A free module is a module that has a basis, or equivalently, one that is isomorphic to a direct sum of copies of the scalar ring

R

.

basis: A basis of a module $M$ is a set of elements in $M$ such that every element in the module can be expressed as a finite sum of elements in the basis in a unique way.

Projective module: A $R$ -module $P$ is called a projective module if given a $R$ -module homomorphism $g: P \to M$ , and a surjective $R$ -module homomorphism $f:N\to M$ , there exists a $R$ -module homomorphism $h : P \to N$ such that $f \circ h = g$ .

The following conditions are equivalent:

The covariant functor $\textrm{Hom}_R(P, - )$ is exact.
$M$ is a projective module.
Every short exact sequence $0 \to L \to L' \to P \to 0$ is split.
$M$ is a direct summand of free modules.

In particular, every free module is projective.

injective module: A $R$ -module $Q$ is called an injective module if given a $R$ -module homomorphism $g: X \to Q$ , and an injective $R$ -module homomorphism $f:X\to Y$ , there exists a

$R$ -module homomorphism $h : Y \to Q$ such that $f \circ h = g$ .

The following conditions are equivalent:

The contravariant functor $\textrm{Hom}_R( - , I)$ is exact.
$I$ is a injective module.
Every short exact sequence $0 \to I \to L \to L' \to 0$ is split.

flat module: A $R$ -module $F$ is called a flat module if the tensor product functor $- \otimes_R F$ is exact.; In particular, every projective module is flat.

simple module: A simple module is a nonzero module whose only submodules are zero and itself.

indecomposable module: An indecomposable module is a non-zero module that cannot be written as a direct sum of two non-zero submodules. Every simple module is indecomposable.

principal indecomposable module: A cyclic indecomposable projective module is known as a PIM.

semisimple module: A module is called semisimple if it is the direct sum of simple submodules.

faithful module: A faithful module $M$ is one where the action of each nonzero $r\in R$ on $M$ is nontrivial (i.e. $rx \ne 0$ for some x in M). Equivalently, $\textrm{Ann}(M)$ is the zero ideal.

Noetherian module: A Noetherian module is a module such that every submodule is finitely generated. Equivalently, every increasing chain of submodules becomes stationary after finitely many steps.

Artinian module: An Artinian module is a module in which every decreasing chain of submodules becomes stationary after finitely many steps.

finite length module: A module which is both Artinian and Noetherian has additional special properties.

graded module: A module $M$ over a graded ring $A = \bigoplus_{n\in \mathbb N}A_n$ is a graded module if $M$ can be expressed as a direct sum $\bigoplus_{i\in \mathbb N}M_i$ and $A_i M_j \subseteq M_{i+j}$ .

invertible module: Roughly synonymous to rank 1 projective module.

uniform module: Module in which every two non-zero submodules have a non-zero intersection.

algebraically compact module (pure injective module): Modules in which all systems of equations can be decided by finitary means. Alternatively, those modules which leave pure-exact sequence exact after applying Hom.

injective cogenerator: An injective module such that every module has a nonzero homomorphism into it.

irreducible module: synonymous to "simple module"

completely reducible module: synonymous to "semisimple module"

Operations on modules

Direct sum of modules

Tensor product of modules

Hom functor

Ext functor

Tor functor

Essential extension: An extension in which every nonzero submodule of the larger module meets the smaller module in a nonzero submodule.

Injective envelope: A maximal essential extension, or a minimal embedding in an injective module

Projective cover: A minimal surjection from a projective module.

Socle: The largest semisimple submodule

Radical of a module: The intersection of the maximal submodules. For Artinian modules, the smallest submodule with semisimple quotient.

Changing scalars

Restriction of scalars: Uses a ring homomorphism from R to S to convert S-modules to R-modules

Extension of scalars: Uses a ring homomorphism from R to S to convert R-modules to S-modules

Localization of a module: Converts R modules to S modules, where S is a localization of R

Endomorphism ring: A left R-module is a right S-module where S is its endomorphism ring.

Homological algebra

Mittag-Leffler condition (ML)
Short five lemma
Five lemma
Snake lemma

Modules over special rings

D-module: A module over a ring of differential operators.
Drinfeld module: A module over a ring of functions on algebraic curve with coefficients from a finite field.
Galois module: A module over the group ring of a Galois group
Structure theorem for finitely generated modules over a principal ideal domain: Finitely generated modules over PIDs are finite direct sums of primary cyclic modules.
Tate module: A special kind of Galois module

Miscellaneous

Rational canonical form
elementary divisor
invariants
fitting ideal
normal forms for matrices
Jordan Hölder composition series
tensor product

References

John A. Beachy (1999). Introductory Lectures on Rings and Modules (1st ed.). Addison-Wesley. ISBN 0-521-64407-0.
Golan, Jonathan S.; Head, Tom (1991), Modules and the structure of rings, Monographs and Textbooks in Pure and Applied Mathematics, 147, Marcel Dekker, ISBN 978-0-8247-8555-0, MR 1201818
Lam, Tsit-Yuen (1999), Lectures on modules and rings, Graduate Texts in Mathematics No. 189, Berlin, New York: Springer-Verlag, ISBN 978-0-387-98428-5, MR 1653294
Serge Lang (1993). Algebra (3rd ed.). Addison-Wesley. ISBN 0-201-55540-9.
Passman, Donald S. (1991), A course in ring theory, The Wadsworth & Brooks/Cole Mathematics Series, Pacific Grove, CA: Wadsworth & Brooks/Cole Advanced Books & Software, ISBN 978-0-534-13776-2, MR 1096302

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