irreducible ideal

• Every prime ideal is irreducible.{{citation|title=Advanced Algebra|first=Anthony W.|last=Knapp|series=Cornerstones|publisher=Springer|year=2007|isbn=9780817645229|page=446|url=https://books.google.com/books?id=25JfJAgqC8sC&pg=PA446}}. Let $J$ and $K$ be ideals of a commutative ring $R$, with neither one contained in the other. Then there exist $a\backslash in\; J\; \backslash setminus\; K$ and $b\backslash in\; K\; \backslash setminus\; J$, where neither is in $J\; \backslash cap\; K$ but the product is. This proves that a reducible ideal is not prime. A concrete example of this are the ideals $2\; \backslash mathbb\; Z$ and $3\; \backslash mathbb\; Z$ contained in $\backslash mathbb\; Z$. The intersection is $6\; \backslash mathbb\; Z$, and $6\; \backslash mathbb\; Z$ is not a prime ideal.
• Every irreducible ideal of a Noetherian ring is a primary ideal, and consequently for Noetherian rings an irreducible decomposition is a primary decomposition.{{cite book|last1=Dummit|first1=David S.|last2=Foote|first2=Richard M.|title=Abstract Algebra|date=2004|publisher=John Wiley & Sons, Inc.|location=Hoboken, NJ|isbn=0-471-43334-9|pages=683–685|edition=Third}}
• Every primary ideal of a principal ideal domain is an irreducible ideal.
• Every irreducible ideal is primal.{{citation

| last = Fuchs | first = Ladislas

| doi = 10.2307/2032421

| journal = Proceedings of the American Mathematical Society

| mr = 0032584

| pages = 1–6

| title = On primal ideals

| volume = 1

| year = 1950| issue = 1

| jstor = 2032421

| doi-access = free

}}. Theorem 1, p. 3.

An element of an integral domain is prime if and only if the ideal generated by it is a non-zero prime ideal. This is not true for irreducible ideals; an irreducible ideal may be generated by an element that is not an irreducible element, as is the case in $\backslash mathbb\; Z$ for the ideal $4\; \backslash mathbb\; Z$ since it is not the intersection of two strictly greater ideals.

In algebraic geometry, if an ideal $I$ of a ring $R$ is irreducible, then $V(I)$ is an irreducible subset in the Zariski topology on the spectrum $\backslash operatorname\{Spec\}\; R$. The converse does not hold; for example the ideal $(x^2,xy,y^2)$ in $\backslash mathbb\; C[x,y]$ defines the irreducible variety consisting of just the origin, but it is not an irreducible ideal as $(x^2,xy,y^2)\; =\; (x^2,y)\; \backslash cap\; (x,y^2)$.

• Irreducible module
• Irreducible space
• Laskerian ring

{{reflist}}

Category:Ring theory

Category:Algebraic topology

{{Abstract-algebra-stub}}