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A special kind of subring of a ring R is the subring generated by a subset X, which is defined as the intersection of all subrings of R containing X. [3] The subring generated by X is also the set of all linear combinations with integer coefficients of elements of X , including the additive identity ("empty combination") and multiplicative ...
An intersection of subrings is a subring. Given a subset E of R, the smallest subring of R containing E is the intersection of all subrings of R containing E, and it is called the subring generated by E. For a ring R, the smallest subring of R is called the characteristic subring of R. It can be generated through addition of copies of 1 and −1.
The left R-module M is finitely generated if there exist a 1, a 2, ..., a n in M such that for any x in M, there exist r 1, r 2, ..., r n in R with x = r 1 a 1 + r 2 a 2 + ... + r n a n. The set {a 1, a 2, ..., a n} is referred to as a generating set of M in this case. A finite generating set need not be a basis, since it need not be linearly ...
For example, we can take the subring of complex numbers of the form +, with and integers. [4] The maximal order question can be examined at a local field level. This technique is applied in algebraic number theory and modular representation theory.
In mathematics, in particular abstract algebra, a graded ring is a ring such that the underlying additive group is a direct sum of abelian groups such that + .The index set is usually the set of nonnegative integers or the set of integers, but can be any monoid.
[2] An (R,S)-bimodule is an abelian group together with both a left scalar multiplication · by elements of R and a right scalar multiplication ∗ by elements of S, making it simultaneously a left R-module and a right S-module, satisfying the additional condition (r · x) ∗ s = r ⋅ (x ∗ s) for all r in R, x in M, and s in S.
One defines the ring of integers of a non-archimedean local field F as the set of all elements of F with absolute value ≤ 1; this is a ring because of the strong triangle inequality. [12] If F is the completion of an algebraic number field, its ring of integers is the completion of the latter's ring of integers. The ring of integers of an ...
For example, M n (Z) is a subring of M n (Q). The matrix ring M n (R) is commutative if and only if n = 0, R = 0, or R is commutative and n = 1. In fact, this is true also for the subring of upper triangular matrices. Here is an example showing two upper triangular 2 × 2 matrices that do not commute, assuming 1 ≠ 0 in R: