Module (mathematics)

In mathematics, a module is one of the fundamental algebraic structures used in abstract algebra. A module over a ring is a generalization of the notion of vector space over a field, where the corresponding scalars are the elements of an arbitrary ring (with identity) and where a multiplication (left and right) is defined. /o to the right) between ring elements and module elements.

Modules are closely related to group representation theory. They are one of the central notions of commutative algebra and homological algebra and are used in algebraic geometry and algebraic topology.

Definition

Sea R{displaystyle R} a ring with identity and 1R{displaystyle 1_{R}} his multipliative identity. A R{displaystyle R}- left hand modedulum M{displaystyle M} It's an Abelian group. (M,+){displaystyle (M,+)} and an operation ⋅ ⋅ :R× × M→ → M{displaystyle cdot:Rtimes Mto M} for any r,s한 한 R{displaystyle r,sin R}, x,and한 한 M{displaystyle x,yin M}You got it.

1. (rs)x=r(sx){displaystyle (rs)x=r(sx)}
2. (r+s)x=rx+sx{displaystyle (r+s)x=rx+sx}
3. r(x+and)=rx+rand{displaystyle r(x+y)=rx+ry}
4. 1x=x{displaystyle 1x=x}

Generally, it is written simply "a R{displaystyle R}-module Left M{displaystyle M}or RM{displaystyle R_{M}}.

Some authors^{[citation required]} omit condition 4 in the general definition of left modules, and call the structures defined above "unital left modules". In this article however, all modules (and all rings) are assumed to be unital. In general, for modules, condition 4 is considered in most texts, while for rings it is not assumed that there is a unit element, unless otherwise stated.

A R{displaystyle R}- Right module M{displaystyle M} or MR{displaystyle M_{R}} is defined similarly, only that the ring acts on the right, that is to say it has a scale multiplication of the form M× × R→ → M{displaystyle Mtimes Rto M}, and the three axioms above are written with the climbers r{displaystyle r} and s{displaystyle s} to the right x{displaystyle x} e and{displaystyle and}.

If R is commutative, then the R-modules on the left are the same as R-modules on the right and are called just R-modules.

Examples

• Yeah. K is a body, then the concepts "K- vectorial space" and K-module are identical.

• Each Abelian group M is a module on the ring of the integers Z if defined nx = x + x +... + x (n sums) for n  0, 0 x = 0, and (- n) x =nx) for n.

• Yeah. R It's any ring. n a natural number, then the Cartesian product Rⁿ is a left and right module on R if component to component operations are used. The case n = 0 da trivial R-module {0} consisting only of the identity element (additional).

• Yeah. X is a differentiable variety, then differential functions X to the real numbers form a ring R. The set of all different vector fields defined in X form a module on R, and the same with the tensoral fields and differential forms in X.

• The matrices squares n- by-n with real inputs form a ring Rand the Euclidean space R ⁿ is a left module on this ring if the module operation is defined via the multiplication of matrices.

• Yeah. R It's any ring. I is any left ideal in R, then I is a left module on R. Similarly, of course, ideal rights are rights modules.

Submodules and homomorphisms

Suppose M is a left R-module and N is a subgroup of M. Then N is a submodule (or R-submodule, to be more explicit) if, for any n in N and any r in R, the product rn is in N (or the nr for a right module). If M and N are R - modules, then a function f: M → N is a homomorphism of R - modules if, for any m, n in M and r, s in R,

f (rm + sn) = rf(m+ sf(n).

This, like any homomorphism of mathematical objects, is precisely a function that preserves the structure of objects. A bijective homomorphism of modules is an isomorphism of modules, and the two modules are called isomorphs. Two isomorphic modules are identical for all practical purposes, differing only in the notation for their elements.

The kernel of a homomorphism of modules f: M → N is the submodule of M consisting of on all elements that are zeroed by f. The familiar isomorphy theorems of abelian groups and of vector spaces are also valid for R-modules.

The left R-modules, together with their module homomorphisms, form a category, written as _RMod. This is an abelian category.

Types of modules

Finitely generated. A module M is finitely generated if there exists a finite number of x₁ elements..., x_n in M such that each element of M is a linear combination of those elements with coefficients of the scalar ring R.

Free. A free module is a module that has a free base, or equivalently, one that is isomorphic to a direct sum of copies of the scalar ring R. These are modules that behave similar to vector spaces.

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

Injective. The injective modules are defined dually to the projective modules.

Simple. A simple module S is a module other than {0} whose only submodules are {0} and S. Simple modules are sometimes called irreducible.

Indecomposable. An indecomposable modulus is a nonzero modulus that cannot be written as a direct sum of two nonzero submodules. Each simple module is indecomposable.

Faithful. A faithful module M is one where the action of each r (non-zero) in R is nontrivial (ie, there exists some m in M such that rm ≠ 0). Equivalently, the nullifier of M is the zero ideal.

Noetherian. A Noetherian module is a module such that each submodule is finitely generated. Equivalently, each increasing chain of submodules becomes stationary in finite steps.

Artinian. An Artinian module is a module in which each decreasing chain of submodules becomes stationary in finite steps.

Alternative definition as representations

If M is a left R-module, then the action of an r element on R is defined as the function M → M that sends each x to the rx (or to xr in the case of a right module), and is necessarily a group endomorphism of the abelian group (M, +). The set of all group endomorphisms of M is denoted End_Z(M) and forms a low ring adding and compounding, and sending an r element of the ring R to your action actually defines a ring homomorphism from R to End _Z(M).

Such a homomorphism R of the ring → End_Z(M) is called a representation of R in the abelian group M; an alternative and equivalent way of defining left R-modules is to say that a left R-module is an abelian group M together with a representation of R on it.

A representation is called true if and only if the function R → End_Z( M) is injective. In modulo terms, this means that if r is an element of R such that rx = 0 for all x in M, then r = 0. Each abelian group is a faithful module over the integers or over some modular arithmetic Z/n Z.

Generalizations

Any R ring can be viewed as a preadditive category with a single object. With this understanding, a left R-module is an additive (covariant) functor from R to the category Ab abelian groups. The right R-modules are contravariant additive functors. This suggests that, if C is any preadditive category, a covariant additive functor from C to Ab is considered a generalized left module over C; these functors form a category of functors C-Mod which is the natural generalization of the category of modules R-Mod.

Modules over commutative rings can be generalized in a different direction: take a ringed space (X, O_X) and consider the bundles of O_X-modules. These form a category O_X-Mod. If X has only one point, then this is a module category in the old sense on the commutative ring O_X(X ).