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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 ...
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.
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.
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.
rngs ⊃ rings ⊃ commutative rings ⊃ integral domains ⊃ integrally closed domains ⊃ GCD domains ⊃ unique factorization domains ⊃ principal ideal domains ⊃ Euclidean domains ⊃ fields ⊃ algebraically closed fields
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 nilradical of a commutative ring is composed entirely of nilpotent elements, and therefore no element can be regular. This gives an example of an ideal which is not a regular element ideal. In an Artinian ring, each element is either invertible or a zero divisor. Because of this, such a ring only has one regular element ideal: just R.
Let A be a simplicial commutative ring. Then the ring structure of A gives = the structure of a graded-commutative graded ring as follows.. By the Dold–Kan correspondence, is the homology of the chain complex corresponding to A; in particular, it is a graded abelian group.