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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 ring is a prime ring if and only if the zero ideal is a prime ideal, and moreover a ring is a domain if and only if the zero ideal is a completely prime ideal. Another fact from commutative theory echoed in noncommutative theory is that if A is a nonzero R - module , and P is a maximal element in the poset of annihilator ideals of submodules ...
The prime spectrum of a Boolean ring (e.g., a power set ring) is a compact totally disconnected Hausdorff space (that is, a Stone space). [4] (M. Hochster) A topological space is homeomorphic to the prime spectrum of a commutative ring (i.e., a spectral space) if and only if it is compact, quasi-separated and sober. [5]
4. prime ring : A nonzero ring R is called a prime ring if for any two elements a and b of R with aRb = 0, we have either a = 0 or b = 0. This is equivalent to saying that the zero ideal is a prime ideal (in the noncommutative sense.) Every simple ring and every domain is a prime ring. primitive 1.
2. A prime element of a ring is an element that generates a prime ideal. 3. A prime local ring is a localization of the integers at a prime ideal. 4. "Prime sequence" is an alternative name for a regular sequence. primary 1. A primary ideal is a proper ideal p of a ring R such that if rm is in p then either m is in p or some power of r is in p.
For example, in the ring of integers Z, (p n) is a primary ideal if p is a prime number. The notion of primary ideals is important in commutative ring theory because every ideal of a Noetherian ring has a primary decomposition , that is, can be written as an intersection of finitely many primary ideals.
For instance, the prime ideals of a ring are analogous to prime numbers, and the Chinese remainder theorem can be generalized to ideals. There is a version of unique prime factorization for the ideals of a Dedekind domain (a type of ring important in number theory ).
An element p of a commutative ring R is said to be prime if it is not the zero element or a unit and whenever p divides ab for some a and b in R, then p divides a or p divides b.With this definition, Euclid's lemma is the assertion that prime numbers are prime elements in the ring of integers.