Module (mathematics)

• If K is a field, then K-vector spaces (vector spaces over K) and K-modules are identical.
• If K is a field, and K[x] a univariate polynomial ring, then a K[x]-module M is a K-module with an additional action of x on M that commutes with the action of K on M. In other words, a K[x]-module is a K-vector space M combined with a linear map from M to M. Applying the Structure theorem for finitely generated modules over a principal ideal domain to this example shows the existence of the rational and Jordan canonical forms.
• The concept of a Z-module agrees with the notion of an abelian group. That is, every abelian group is a module over the ring of integers Z in a unique way. For n > 0, let n ⋅ x = x + x + ... + x (n summands), 0 ⋅ x = 0, and (−n) ⋅ x = −(n ⋅ x). Such a module need not have a basis—groups containing torsion elements do not. (For example, in the group of integers modulo 3, one cannot find even one element which satisfies the definition of a linearly independent set since when an integer such as 3 or 6 multiplies an element, the result is 0. However, if a finite field is considered as a module over the same finite field taken as a ring, it is a vector space and does have a basis.)
• The decimal fractions (including negative ones) form a module over the integers. Only singletons are linearly independent sets, but there is no singleton that can serve as a basis, so the module has no basis and no rank.
• If R is any ring and n a natural number, then the cartesian product Rⁿ is both a left and right R-module over R if we use the component-wise operations. Hence when n = 1, R is an R-module, where the scalar multiplication is just ring multiplication. The case n = 0 yields the trivial R-module {0} consisting only of its identity element. Modules of this type are called free and if R has invariant basis number (e.g. any commutative ring or field) the number n is then the rank of the free module.
• If M_n(R) is the ring of n × n matrices over a ring R, M is an M_n(R)-module, and e_i is the n × n matrix with 1 in the (i, i)-entry (and zeros elsewhere), then e_iM is an R-module, since re_im = e_irm ∈ e_iM. So M breaks up as the direct sum of R-modules, M = e₁M ⊕ ... ⊕ e_nM. Conversely, given an R-module M₀, then M₀^⊕n is an M_n(R)-module. In fact, the category of R-modules and the category of M_n(R)-modules are equivalent. The special case is that the module M is just R as a module over itself, then Rⁿ is an M_n(R)-module.
• If S is a nonempty set, M is a left R-module, and M^S is the collection of all functions f : S → M, then with addition and scalar multiplication in M^S defined pointwise by (f + g)(s) = f(s) + g(s) and (rf)(s) = rf(s), M^S is a left R-module. The right R-module case is analogous. In particular, if R is commutative then the collection of R-module homomorphisms h : M → N (see below) is an R-module (and in fact a submodule of N^M).
• If X is a smooth manifold, then the smooth functions from X to the real numbers form a ring C^∞(X). The set of all smooth vector fields defined on X form a module over C^∞(X), and so do the tensor fields and the differential forms on X. More generally, the sections of any vector bundle form a projective module over C^∞(X), and by Swan's theorem, every projective module is isomorphic to the module of sections of some bundle; the category of C^∞(X)-modules and the category of vector bundles over X are equivalent.
• If R is any ring and I is any left ideal in R, then I is a left R-module, and analogously right ideals in R are right R-modules.
• If R is a ring, we can define the opposite ring R^op which has the same underlying set and the same addition operation, but the opposite multiplication: if ab = c in R, then ba = c in R^op. Any left R-module M can then be seen to be a right module over R^op, and any right module over R can be considered a left module over R^op.
• Modules over a Lie algebra are (associative algebra) modules over its universal enveloping algebra.
• If R and S are rings with a ring homomorphism φ : R → S, then every S-module M is an R-module by defining rm = φ(r)m. In particular, S itself is such an R-module.

Suppose M is a left R-module and N is a subgroup of M. Then N is a submodule (or more explicitly an R-submodule) if for any n in N and any r in R, the product r ⋅ n (or n ⋅ r for a right R-module) is in N.

If X is any subset of an R-module, then the submodule spanned by X is defined to be ${\textstyle \langle X\rangle =\,\bigcap _{N\supseteq X}N}$ where N runs over the submodules of M which contain X, or explicitly ${\textstyle \left\{\sum _{i=1}^{k}r_{i}x_{i}\mid r_{i}\in R,x_{i}\in X\right\}}$, which is important in the definition of tensor products.[2]

The set of submodules of a given module M, together with the two binary operations + and ∩, forms a lattice which satisfies the modular law: Given submodules U, N₁, N₂ of M such that N₁ ⊂ N₂, then the following two submodules are equal: (N₁ + U) ∩ N₂ = N₁ + (U ∩ N₂).

If M and N are left R-modules, then a map f : M → N is a homomorphism of R-modules if for any m, n in M and r, s in R,

${\displaystyle f(r\cdot m+s\cdot n)=r\cdot f(m)+s\cdot f(n)}$.

This, like any homomorphism of mathematical objects, is just a mapping which preserves the structure of the objects. Another name for a homomorphism of R-modules is an R-linear map.

A bijective module homomorphism f : M → N is called a module isomorphism, and the two modules M and N are called isomorphic. Two isomorphic modules are identical for all practical purposes, differing solely in the notation for their elements.

The kernel of a module homomorphism f : M → N is the submodule of M consisting of all elements that are sent to zero by f, and the image of f is the submodule of N consisting of values f(m) for all elements m of M.[3] The isomorphism theorems familiar from groups and vector spaces are also valid for R-modules.

Given a ring R, the set of all left R-modules together with their module homomorphisms forms an abelian category, denoted by R-Mod (see category of modules).

Finitely generated

An R-module M is finitely generated if there exist finitely many elements x₁, ..., x_n in M such that every element of M is a linear combination of those elements with coefficients from the ring R.

Cyclic

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

Free

A free R-module is a module that has a basis, or equivalently, one that is isomorphic to a direct sum of copies of the ring R. These are the modules that behave very much like vector spaces.

Projective

Projective modules are direct summands of free modules and share many of their desirable properties.

Injective

Injective modules are defined dually to projective modules.

Flat

A module is called flat if taking the tensor product of it with any exact sequence of R-modules preserves exactness.

Torsionless

A module is called torsionless if it embeds into its algebraic dual.

Simple

A simple module S is a module that is not {0} and whose only submodules are {0} and S. Simple modules are sometimes called irreducible.[4]

Semisimple

A semisimple module is a direct sum (finite or not) of simple modules. Historically these modules are also called completely reducible.

Indecomposable

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, but there are indecomposable modules which are not simple (e.g. uniform modules).

Faithful

A faithful module M is one where the action of each r ≠ 0 in R on M is nontrivial (i.e. r ⋅ x ≠ 0 for some x in M). Equivalently, the annihilator of M is the zero ideal.

Torsion-free

A torsion-free module is a module over a ring such that 0 is the only element annihilated by a regular element (non zero-divisor) of the ring, equivalently ${\displaystyle rm=0}$ implies ${\displaystyle r=0}$ or ${\displaystyle m=0}$.

Noetherian

A Noetherian module is a module which satisfies the ascending chain condition on submodules, that is, every increasing chain of submodules becomes stationary after finitely many steps. Equivalently, every submodule is finitely generated.

Artinian

An Artinian module is a module which satisfies the descending chain condition on submodules, that is, every decreasing chain of submodules becomes stationary after finitely many steps.

Graded

A graded module is a module with a decomposition as a direct sum M = ⨁_x M_x over a graded ring R = ⨁_x R_x such that R_xM_y ⊂ M_x+y for all x and y.

Uniform

A uniform module is a module in which all pairs of nonzero submodules have nonzero intersection.

• Group ring
• Algebra (ring theory)
• Module (model theory)
• Module spectrum
• Annihilator

• F.W. Anderson and K.R. Fuller: Rings and Categories of Modules, Graduate Texts in Mathematics, Vol. 13, 2nd Ed., Springer-Verlag, New York, 1992, ISBN 0-387-97845-3, ISBN 3-540-97845-3
• Nathan Jacobson. Structure of rings. Colloquium publications, Vol. 37, 2nd Ed., AMS Bookstore, 1964, ISBN 978-0-8218-1037-8

• "Module", Encyclopedia of Mathematics, EMS Press, 2001 [1994]
• Why is it a good idea to study the modules of a ring ? on MathOverflow
• module in nLab