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The cohomology of a cdga is a graded-commutative ring, sometimes referred to as the cohomology ring. 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.
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.
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.
The classical ring of quotients for any commutative Noetherian ring is a semilocal ring. The endomorphism ring of an Artinian module is a semilocal ring. Semi-local rings occur for example in algebraic geometry when a (commutative) ring R is localized with respect to the multiplicatively closed subset S = ∩ (R \ p i ) , where the p i are ...
The intersection of members of a descending chain of prime ideals is a prime ideal, and in a commutative ring the union of members of an ascending chain of prime ideals is a prime ideal. With Zorn's Lemma, these observations imply that the poset of prime ideals of a commutative ring (partially ordered by inclusion) has maximal and minimal elements.
In mathematics, rings are algebraic structures that generalize fields: multiplication need not be commutative and multiplicative inverses need not exist. Informally, a ring is a set equipped with two binary operations satisfying properties analogous to those of addition and multiplication of integers.
For example, a polynomial ring R[x] is finitely generated by {1, x} as a ring, but not as a module. If A is a commutative algebra (with unity) over R, then the following two statements are equivalent: [5] A is a finitely generated R module. A is both a finitely generated ring over R and an integral extension of R.
Commutative rings are much better understood than noncommutative ones. Algebraic geometry and algebraic number theory, which provide many natural examples of commutative rings, have driven much of the development of commutative ring theory, which is now, under the name of commutative algebra, a major area of modern mathematics. Because these ...