In abstract algebra, a module is defined to be indecomposable if it is non-zero and cannot be written as a direct sum of two non-zero submodules. In many situations, all modules of interest can be written as direct sums of indecomposable ones; the indecomposable modules can then be thought of as the "basic building blocks", the only objects that need to be studied. Abstract algebra is the field of mathematics concerned with the study of algebraic structures such as groups, rings and fields. ... In abstract algebra, a module is a generalization of a vector space. ... In abstract algebra, the direct sum is a construction which combines several vector spaces (or groups, or abelian groups, or modules) into a new, bigger one. ... In abstract algebra, a module is a generalization of a vector space. ...

Examples

The modules over fields are vector spaces. A vector space is indecomposable if and only if its dimension is 1. So every vector space is a direct sum of indecomposable ones (with infinitely many summands if the dimension is infinite). In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... A vector space (or linear space) is the basic object of study in the branch of mathematics called linear algebra. ... In mathematics, the dimension of a vector space V is the cardinality (i. ...

The modules over the ring of integers Z are the abelian groups. An abelian group is indecomposable if and only if it is isomorphic to Z or to a factor group of the form Z/pⁿZ for some prime number p and some positive integer n. Every finitely-generated abelian group is a direct sum of (finitely many) indecomposable abelian groups. There are, however, abelian groups that cannot be written as a (finite or infinite) direct sum of indecomposables; the rational numbers Q form the simplest example. The integers consist of the positive natural numbers (1, 2, 3, …), their negatives (−1, −2, −3, ...) and the number zero. ... In mathematics, an abelian group, also called a commutative group, is a group (G, *) such that a * b = b * a for all a and b in G. Abelian groups are named after Niels Henrik Abel. ... In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of interesting mapping between objects. ... In mathematics, given a group G and a normal subgroup N of G, the quotient group, or factor group, of G over N is a group that intuitively collapses the normal subgroup N to the identity element. ... In mathematics, a prime number (or prime) is a natural number greater than one whose only positive divisors are one and itself. ... In abstract algebra, an abelian group (G,+) is called finitely generated if there exist finitely many elements x1,...,xs in G such that every x in G can be written in the form x = n1x1 + n2x2 + ... + nsxs with integers n1,...,ns. ... In mathematics, a rational number (or informally fraction) is a ratio or quotient of two integers, usually written as the vulgar fraction a/b, where b is not zero. ...

The above situation over Z can be generalized in a straight-forward manner to modules over any principal ideal domain R: the indecomposable modules are R and R/pⁿR for prime ideals p in R. Every finitely-generated R-module is a direct sum of these. In abstract algebra, a principal ideal domain (PID) is an integral domain in which every ideal is principal (that is, generated by a single element). ... In mathematics, a prime ideal is a subset of a ring which shares many important properties of a prime number in the ring of integers. ...

For a fixed positive integer n, consider the ring R of n-by-n matrices with entries from the real numbers (or from any other field K). Then Kⁿ is a left R-module (the scalar multiplication is matrix multiplication). This is up to isomorphism the only indecomposable module over R. Every left R-module is a direct sum of (finitely or infinitely many) copies of this module Kⁿ. For the square matrix section, see square matrix. ... In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ... This article gives an overview of the various ways to multiply matrices. ... In mathematics, the term up to xxxx is used to describe a situation in which members of an equivalence class can be regarded as a single entity for some purpose. ...

Facts

Every simple module is indecomposable. The converse is not true in general, as is shown by the second example above. In abstract algebra, a (left or right) module S over a ring R is called simple if it is not the zero module and if its only submodules are 0 and S. Understanding the simple modules over a ring is usually helpful because they form the building blocks of all...

By looking at the endomorphism ring of a module, one can tell whether the module is indecomposable: if and only if the endomorphism ring does not contain an idempotent different from 0 and 1. (If f is such an idempotent endomorphism of M, then M is the direct sum of ker(f) and im(f).) In abstract algebra, one associates to certain objects a ring, the objects endomorphism ring, which encodes several internal properties of the object. ... In mathematics, an idempotent element is an element which, intuitively, leaves something unchanged. ...

A module of finite length is indecomposable if and only if its endomorphism ring is local. Still more information about endomorphisms of finite-length indecomposables is provided by the Fitting lemma. In abstract algebra, the length of a module is a measure of the modules size. It is defined as the length of the longest ascending chain of submodules and is a generalization of the concept of dimension for vector spaces. ... In mathematics, more particularly in abstract algebra, local rings are certain rings that are comparatively simple, and serve to describe the local behavior of functions defined on varieties or manifolds. ... The Fitting lemma, named after the mathematician Hans Fitting, is a basic statement in abstract algebra. ...

In the finite-length situation, decomposition into indecomposables is particularly useful, because of the Krull-Schmidt theorem: every finite-length module can be written as a direct sum of finitely many indecomposable modules, and this decomposition is essentially unique (meaning that if you have a different decomposition into indecomposable, then the summands of the first decomposition can be paired off with the summands of the second decomposition so that the members of each pair are isomorphic). The Krull-Schmidt theorem states that a group , subjected to certain finiteness conditions of chains of subgroups, can be uniquely written as a finite product of indecomposable subgroups. ...