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Ring theory studies the structure of rings; their representations, or, in different language, modules; special classes of rings (group rings, division rings, universal enveloping algebras); related structures like rngs; as well as an array of properties that prove to be of interest both within the theory itself and for its applications, such as ...
In mathematics, and more specifically in ring theory, Krull's theorem, named after Wolfgang Krull, asserts that a nonzero ring [1] has at least one maximal ideal.The theorem was proved in 1929 by Krull, who used transfinite induction.
In algebra, a commutative Noetherian ring A is said to have the approximation property with respect to an ideal I if each finite system of polynomial equations with coefficients in A has a solution in A if and only if it has a solution in the I-adic completion of A. [1] [2] The notion of the approximation property is due to Michael Artin.
A ring R is said to act densely on a simple right R-module U if it satisfies the conclusion of the Jacobson density theorem. [7] There is a topological reason for describing R as "dense". Firstly, R can be identified with a subring of End( D U ) by identifying each element of R with the D linear transformation it induces by right multiplication.
Some examples of orders are: [2] If is the matrix ring over , then the matrix ring () over is an -order in ; If is an integral domain and a finite separable extension of , then the integral closure of in is an -order in .
In mathematics, more specifically ring theory, a left, right or two-sided ideal of a ring is said to be a nil ideal if each of its elements is nilpotent. [1] [2]The nilradical of a commutative ring is an example of a nil ideal; in fact, it is the ideal of the ring maximal with respect to the property of being nil.
Let R be a ring, [a] and let a and b be elements of R. If there exists an element x in R with ax = b , one says that a is a left divisor of b and that b is a right multiple of a . [ 1 ] Similarly, if there exists an element y in R with ya = b , one says that a is a right divisor of b and that b is a left multiple of a .
In mathematics, specifically ring theory, a left primitive ideal is the annihilator of a (nonzero) simple left module. A right primitive ideal is defined similarly. Left and right primitive ideals are always two-sided ideals. Primitive ideals are prime. The quotient of a ring by a left primitive ideal is a left primitive ring.