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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 ...
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.
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 is a set R equipped with two binary operations [a] + (addition) and ⋅ (multiplication) satisfying the following three sets of axioms, called the ring axioms: [1] [2] [3] R is an abelian group under addition, meaning that: (a + b) + c = a + (b + c) for all a, b, c in R (that is, + is associative). a + b = b + a for all a, b in R (that ...
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 ...
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.
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 .
Let R be a ring, and let I be a proper ideal of R. Then there is a maximal ideal of R containing I. The statement of the original theorem can be obtained by taking I to be the zero ideal (0). Conversely, applying the original theorem to R/I leads to this result.