Intersections of essential minimal prime ideals

Let $\mathcal{Z(R)}$ be the set of zero divisor elements of a commutative ring $R$ with identity and $\mathcal{M}$ be the space of minimal prime ideals of $R$ with Zariski topology. An ideal $I$ of $R$ is called strongly dense ideal or briefly $sd$-ideal if $I\subseteq \mathcal{Z(R)}$ and is contained in no minimal prime ideal. We denote by $R_{K}(\mathcal{M})$, the set of all $a\in R$ for which $\bar{D(a)}=\bar{\mathcal{M}\setminus V(a)}$ is compact. We show that $R$ has property $(A)$ and $\mathcal{M}$ is compact \ifif $R$ has no $sd$-ideal. It is proved that $R_{K}(\mathcal{M})$ is an essential ideal (resp., $sd$-ideal) \ifif $\mathcal{M}$ is an almost locally compact (resp., $\mathcal{M}$ is a locally compact non-compact) space. The intersection of essential minimal prime ideals of a reduced ring $R$ need not be an essential ideal. We find an equivalent condition for which any (resp., any countable) intersection of essential minimal prime ideals of a reduced ring $R$ is an essential ideal. Also it is proved that the intersection of essential minimal prime ideals of $C(X)$ is equal to the socle of C(X) (i.e., $C_{F}(X)=O^{\beta X\setminus I(X)}$). Finally, we show that a topological space $X$ is pseudo-discrete \ifif $I(X)=X_{L}$ and $C_{K}(X)$ is a pure ideal.Comment: 9 pages, accepted for publication in in Commentationes Mathematicae Universitatis Carolina

Ideals of Adjacent Minors

We give a description of the minimal primes of the ideal generated by the 2 x 2 adjacent minors of a generic matrix. We also compute the complete prime decomposition of the ideal of adjacent m x m minors of an m x n generic matrix when the characteristic of the ground field is zero. A key intermediate result is the proof that the ideals which appear as minimal primes are, in fact, prime ideals. This introduces a large new class of mixed determinantal ideals that are prime

Prime splittings of Determinantal Ideals

We consider determinantal ideals, where the generating minors are encoded in a hypergraph. We study when the generating minors form a Gr\"obner basis. In this case, the ideal is radical, and we can describe algebraic and numerical invariants of these ideals in terms of combinatorial data of their hypergraphs, such as the clique decomposition. In particular, we can construct a minimal free resolution as a tensor product of the minimal free resolution of their cliques. For several classes of hypergraphs we find a combinatorial description of the minimal primes in terms of a prime splitting. That is, we write the determinantal ideal as a sum of smaller determinantal ideals such that each minimal prime is a sum of minimal primes of the summands.Comment: Final version to appear in Communications in Algebr

Minimal prime ideals of skew Hurwitz series ring

Let R be a ring with an endomorphism α. In this paper we obtain necessary and sufficient conditions on R and α such that the skew Hurwitz series ring (HR, α) is a 2-primal ring. In particular, it is proved that, under suitable conditions, (HR,α) is 2-primal if and only if for every minimal prime ideal P∗ in (HR, α) there exists a minimal prime ideal P of R such that P is completely prime and P∗=(HP,α) if and only if P((HR,α))=(H(nil(R)),α) if and only if R is 2-primal and nil((HR,α))=(H(nil(R)),α) if and only if every minimal α-prime ideal of R is completely prime