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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 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.
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.
A subdomain of a division ring which is not right or left Ore: If F is any field, and = , is the free monoid on two symbols x and y, then the monoid ring [] does not satisfy any Ore condition, but it is a free ideal ring and thus indeed a subring of a division ring, by (Cohn 1995, Cor 4.5.9).
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.
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 .
The center of the (full) matrix ring with entries in a commutative ring R consists of R-scalar multiples of the identity matrix. [1] Let F be a field extension of a field k, and R an algebra over k. Then Z(R ⊗ k F) = Z(R) ⊗ k F. The center of the universal enveloping algebra of a Lie algebra plays an important role in the representation ...
Let R be a ring that is graded by the ordered semigroup of non-negative integers, and let + denote the ideal generated by positively graded elements. Then if M is a graded module over R for which M i = 0 {\displaystyle M_{i}=0} for i sufficiently negative (in particular, if M is finitely generated and R does not contain elements of negative ...