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Every module over a division ring is a free module (has a basis); consequently, much of linear algebra can be carried out over a division ring instead of a field. The study of conjugacy classes figures prominently in the classical theory of division rings; see, for example, the Cartan–Brauer–Hua theorem.
Noncommutative rings and associative algebras (rings that are also vector spaces) are often studied via their categories of modules. A module over a ring is an abelian group that the ring acts on as a ring of endomorphisms , very much akin to the way fields (integral domains in which every non-zero element is invertible) act on vector spaces.
The ring of formal power series over the complex numbers is a UFD, but the subring of those that converge everywhere, in other words the ring of entire functions in a single complex variable, is not a UFD, since there exist entire functions with an infinity of zeros, and thus an infinity of irreducible factors, while a UFD factorization must be ...
Rings • Subrings • Ideal • Quotient ring • Fractional ideal • Total ring of fractions • Product of rings • Free product of associative algebras • Tensor product of algebras. Ring homomorphisms • Kernel • Inner automorphism • Frobenius endomorphism. Algebraic structures • Module • Associative algebra • Graded ring ...
Let be a group, written multiplicatively, and let be a ring. The group ring of over , which we will denote by [], or simply , is the set of mappings : of finite support (() is nonzero for only finitely many elements ), where the module scalar product of a scalar in and a mapping is defined as the mapping (), and the module group sum of two mappings and is defined as the mapping () + ().
For example, a polynomial ring R[x] is finitely generated by {1, x} as a ring, but not as a module. If A is a commutative algebra (with unity) over R, then the following two statements are equivalent: [5] A is a finitely generated R module. A is both a finitely generated ring over R and an integral extension of R.
The isomorphism theorems for vector spaces (modules over a field) and abelian groups (modules over ) are special cases of these. For finite-dimensional vector spaces, all of these theorems follow from the rank–nullity theorem. In the following, "module" will mean "R-module" for some fixed ring R.
The category K-Vect (some authors use Vect K) has all vector spaces over a field K as objects, and K-linear maps as morphisms. Since vector spaces over K (as a field) are the same thing as modules over the ring K, K-Vect is a special case of R-Mod (some authors use Mod R), the category of left R-modules. Much of linear algebra concerns the ...