Prime ideal - Wikipedia
WEBIn algebra, a prime ideal is a subset of a ring that shares many important properties of a prime number in the ring of integers. [1][2] The prime ideals for the integers are the sets that contain all the multiples of a given prime number, together with the zero ideal.
Prime Ideal -- from Wolfram MathWorld
WEBAug 22, 2024 · A prime ideal is an ideal I such that if ab in I, then either a in I or b in I. For example, in the integers, the ideal a=<p> (i.e., the multiples of p) is prime whenever p is a prime number. In any principal ideal domain, prime ideals are generated by prime elements.
Prime ideals: definition, verification, and examples
WEBNov 18, 2023 · For an example of a prime ideal, consider $(x)$ in the polynomial ring $k[x,y]$ (all polynomials in $x,y$ over a field $k$). The quotient $k[x,y]/(x)$ is isomorphic to $k[y]$ which is an integral domain, hence $(x)$ is prime. However, the intersection of the two prime ideals $(x)$ and $(y)$, $(x) \cap (y)$ is $(xy)$, which is not prime.
WEBPrime Ideals. Throughout this lecture, we assume that R is a commutative ring with identity. rime Ideals. Every proper ideal of R is contained in a maximal ideal ( orollary 2). To argue such a result, one needs to appeal to Zorn’s lemma, which …
8.4: Maximal and Prime Ideals - Mathematics LibreTexts
WEBApr 17, 2022 · Definition: Prime Ideal. Assume \(R\) is a commutative ring with 1. An ideal \(P\) is called a prime ideal if \(P\neq R\) and whenever the product \(ab\in P\) for \(a,b\in R\), then at least one of \(a\) or \(b\) is in \(P\).
Prime ideal - Encyclopedia of Mathematics
WEBSep 18, 2016 · Prime ideal. A two-sided ideal $I$ of a ring $A$ such that the inclusion $PQ\subseteq I$ for any two-sided ideals $P$ and $Q$ of $A$ implies that either …
WEBLet R be an integral domain and let a be a non-zero element of R. We say that a is prime, if (a) is a prime ideal, not equal to the whole of R. Note that the condition that (a) is not the whole of R is equivalent to requiring that a is not a unit. Definition 18.4.
9 - Ideals and Prime Factorization - Cambridge University Press ...
WEBThe remaining step on the road to unique prime ideal factorization is to define "prime ideal" itself. This involves a definition of "division" for ideals. If we also define "product" of ideals, then a prime ideal is one with a property originally discovered by Euclid: if a prime p divides a product ab, then p divides a or p divides b.
16.6: Maximal and Prime Ideals - Mathematics LibreTexts
WEBA proper ideal \(P\) in a commutative ring \(R\) is called a prime ideal if whenever \(ab \in P\text{,}\) then either \(a \in P\) or \(b \in P\text{.}\) 6
prime ideal in nLab - ncatlab.org
WEBSep 4, 2018 · Idea. Prime ideals are supposed to be a generalization of prime numbers from elements of the ring of integers to ideals in the sense of ‘ideal elements’ of an arbitrary ring (usually commutative, but also possibly something more general than a ring).It's not clear that they do so; maximal ideals may do a better job. (In particular, zero is not a prime number, and the zero ideal of ℤ ...