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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 ...
For example, an exterior algebra is generally not a commutative ring but is a graded-commutative ring. A cup product on cohomology satisfies the skew-commutative relation; hence, a cohomology ring is graded-commutative. In fact, many examples of graded-commutative rings come from algebraic topology and homological algebra.
In commutative and homological algebra, the grade of a finitely generated module over a Noetherian ring is a cohomological invariant defined by vanishing of Ext-modules [1] = = {: (,)}.
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.
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.
Equivalently, every finite division ring is commutative. K is the function field of an algebraic curve over an algebraically closed field (Tsen's theorem). [3] More generally, the Brauer group vanishes for any C 1 field. K is an algebraic extension of Q containing all roots of unity. [2] The Brauer group Br R of the real numbers is the cyclic ...
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.
If the ring R is a ring with no nilpotent elements, then the Frobenius endomorphism is injective: F(r) = 0 means r p = 0, which by definition means that r is nilpotent of order at most p. In fact, this is necessary and sufficient, because if r is any nilpotent, then one of its powers will be nilpotent of order at most p .
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