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In Ring, the category of rings with unity and unity-preserving morphisms, the ring of integers Z is an initial object. The zero ring consisting only of a single element 0 = 1 is a terminal object. In Rig, the category of rigs with unity and unity-preserving morphisms, the rig of natural numbers N is an initial object.
A broad range examples of graded rings arises in this way. For example, the Lazard ring is the ring of cobordism classes of complex manifolds. A graded-commutative ring with respect to a grading by Z/2 (as opposed to Z) is called a superalgebra. A related notion is an almost commutative ring, which means that R is filtered in such a way that ...
A commutative ring (not necessarily a domain) with unity satisfying this condition is called a containment-division ring (CDR). [2] Thus a Dedekind domain is a domain that either is a field, or satisfies any one, and hence all five, of (DD1) through (DD5). Which of these conditions one takes as the definition is therefore merely a matter of taste.
The unit group of the ring M n (R) of n × n matrices over a ring R is the group GL n (R) of invertible matrices. For a commutative ring R, an element A of M n (R) is invertible if and only if the determinant of A is invertible in R. In that case, A −1 can be given explicitly in terms of the adjugate matrix.
As an example, the nilradical of a ring, the set of all nilpotent elements, is not necessarily an ideal unless the ring is commutative. Specifically, the set of all nilpotent elements in the ring of all n × n matrices over a division ring never forms an ideal, irrespective of the division ring chosen.
Prominent examples of commutative rings include polynomial rings; rings of algebraic integers, including the ordinary integers; and p-adic integers. [ 1 ] Commutative algebra is the main technical tool of algebraic geometry , and many results and concepts of commutative algebra are strongly related with geometrical concepts.
the word "example" appears no more than three times in this article, yet examples of rings are what makes ring theory important. 西米露喜欢橙子 21:33, 4 November 2023 (UTC) The right place for examples of rings is in Ring (mathematics), an article linked to in the first sentence of this article. This article contains many examples that ...
In a commutative ring with unity, every maximal ideal is a prime ideal. The converse is not always true: for example, in any nonfield integral domain the zero ideal is a prime ideal which is not maximal. Commutative rings in which prime ideals are maximal are known as zero-dimensional rings, where the dimension used is the Krull dimension.