simple module

In mathematics, specifically in ring theory, the simple modules over a ring R are the (left or right) modules over R that are non-zero and have no non-zero proper submodules. Equivalently, a module M is simple if and only if every cyclic submodule generated by a element of M equals M. Simple modules form building blocks for the modules of finite length, and they are analogous to the simple groups in group theory. In this article, all modules will be assumed to be right unital modules over a ring R.

In mathematics, specifically in ring theory, the simple modules over a ring R are the (left or right) modules over R that are non-zero and have no non-zero proper submodules. Equivalently, a module M is simple if and only if every cyclic submodule generated by a element of M equals M. Simple modules form building blocks for the modules of finite length, and they are analogous to the simple groups in group theory. In this article, all modules will be assumed to be right unital modules over a ring R. Every simple R-module is isomorphic to a quotient R/m where m is a maximal right ideal of R. By the above paragraph, any quotient R/m is a simple module. Conversely, suppose that M is a simple R-module. Then, for any non-zero element x of M, the cyclic submodule xR must equal M. Fix such an x. The statement that xR = M is equivalent to the surjectivity of the homomorphism that sends r to xr. The kernel of this homomorphism is a right ideal I of R, and a standard theorem states that M is isomorphic to R/I. By the above paragraph, we find that I is a maximal right ideal. Therefore, M is isomorphic to a quotient of R by a maximal right ideal. If k is a field and G is a group, then a group representation of G is a left module over the group ring k[G] (for details, see the main article on this relationship). The simple k[G]-modules are also known as irreducible representations. A major aim of representation theory is to understand the irreducible representations of groups. Let M and N be (left or right) modules over the same ring, and let be a module homomorphism. If M is simple, then f is either the zero homomorphism or injective because the kernel of f is a submodule of M. If N is simple, then f is either the zero homomorphism or surjective because the image of f is a submodule of N. If M = N , then f is an endomorphism of M, and if M is simple, then the prior two statements imply that f is either the zero homomorphism or an isomorphism. Consequently, the endomorphism ring of any simple module is a division ring. This result is known as Schur's lemma. The converse of Schur's lemma is not true in general. For example, the Z-module Q is not simple, but its endomorphism ring is isomorphic to the field Q. Let U be a simple right R-module and let D = End R (U). Let A be any D-linear operator on U and let X be a finite D-linearly independent subset of U. Then there exists an element r of R such that x⋅A = x⋅r for all x in X. In particular, any primitive ring may be viewed as (that is, isomorphic to) a ring of D-linear operators on some D-space. A consequence of the Jacobson density theorem is Wedderburn's theorem; namely that any right Artinian simple ring is isomorphic to a full matrix ring of n-by-n matrices over a division ring for some n. This can also be established as a corollary of the Artin–Wedderburn theorem. ==See also== Semisimple modules are modules that can be written as a sum of simple submodules Irreducible ideal Irreducible representation ==References==

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