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A ring R is prime if and only if the zero ideal {0} is a prime ideal in the noncommutative sense. This being the case, the equivalent conditions for prime ideals yield the following equivalent conditions for R to be a prime ring: For any two ideals A and B of R, AB = {0} implies A = {0} or B = {0}.
A (meet-)irreducible ring is a ring in which the intersection of two non-zero ideals is always non-zero. A directly irreducible ring is a ring which cannot be written as the direct sum of two non-zero rings. A subdirectly irreducible ring is a ring with a unique, non-zero minimum two-sided ideal.
Another important example of a DVR is the ring of formal power series = [[]] in one variable over some field .The "unique" irreducible element is , the maximal ideal of is the principal ideal generated by , and the valuation assigns to each power series the index (i.e. degree) of the first non-zero coefficient.
The ring = of algebraic integers in a number field K is Noetherian, integrally closed, and of dimension one: to see the last property, observe that for any nonzero prime ideal I of R, R/I is a finite set, and recall that a finite integral domain is a field; so by (DD4) R is a Dedekind domain. As above, this includes all the examples considered ...
In a commutative ring R with at least two elements, if every proper ideal is prime, then the ring is a field. (If the ideal (0) is prime, then the ring R is an integral domain. If q is any non-zero element of R and the ideal (q 2) is prime, then it contains q and then q is invertible.)
Z[ω] (where ω is a primitive (non-real) cube root of unity), the ring of Eisenstein integers. Define f (a + bω) = a 2 − ab + b 2, the norm of the Eisenstein integer a + bω. K[X], the ring of polynomials over a field K. For each nonzero polynomial P, define f (P) to be the degree of P. [4] K[[X]], the ring of formal power series over the ...