Let $k$ be a field and $R$ a pure subring of the infinite-dimensional polynomial ring $k[X_1,\ldots ]$. If $R$ is generated by monomials, then we show that the equality of height and grade holds for all ideals of~$R$. Also, we show $R$ satisfies the weak Bourbaki unmixed property. As an application, we give the Cohen-Macaulay property of the invariant ring of the action of a linearly reductive group acting by $k$-automorphism on $k[X_1,\ldots ]$. This provides several examples of non Noetherian Cohen-Macaulay rings (e.g., Veronese, determinantal and Grassmanian rings).

We give a general upper bound for the arithmetical rank of the ideals generated by the 2-minors of scroll matrices with entries in an arbitrary commutative unit ring.

For a UMT-domain $D$, we characterize when the polynomial ring $D[X]$ is $t$-compactly packed and every prime $t$-ideal of $D[X]$ is radically perfect. As a corollary, for a quasi-Pr\"ufer domain $D$, we also characterize when every prime ideal of $D[X]$ is radically perfect. Finally we introduce the concepts of Serre's conditions in strong Mori domains and characterize Krull domains and almost factorial domains, respectively.

There are many examples of the fact that dimension and codimension behave somewhat counterintuitively. In \cite {EGAIV1}, it is stated that a topological space is equidimensional, equicodimensional and catenary if and only if every maximal chain of irreducible closed subsets has the same length. We construct examples that show that this is not even true for the spectrum of a Noetherian ring. This gives rise to two notions of biequidimensionality, and we show how these relate to the dimension formula and the existence of a codimension function.

Let $f: A\rightarrow B$ and $g: A\rightarrow C$ be two commutative ring homomorphisms, and let $J$ and $J'$ be two ideals of $B$ and $C$, respectively, such that $f^{-1}(J)=g^{-1}(J')$. The \textit {bi-amalgamation} of $A$ with $(B, C)$ along $(J, J')$ with respect to $(f,g)$ is the subring of $B\times C$ given by \[ A\bowtie ^{f,g}(J,J'):=\big \{(f(a)+j,g(a)+j') \mid a\in A, (j,j')\in J\times J'\big \}. \] In this paper, we investigate ring-theoretic properties of \textit {bi-amalgamations} and capitalize on previous work carried out on various settings of pullbacks and amalgamations. In the second and third sections, we provide examples of bi-amalgamations and show how these constructions arise as pullbacks. The fourth section investigates the transfer of some basic ring theoretic properties to bi-amalgamations, and the fifth section is devoted to the prime ideal structure of these constructions. All new results agree with recent studies in the literature on D'Anna, Finocchiaro and Fontana's amalgamations and duplications.

Let $R$ be a polynomial ring over a field~$K$. To a given squarefree monomial ideal $I \subset R$, one can associate a hypergraph $\mathcal{H} (I)$. In this article, we prove that the arithmetical rank of $I$ is equal to the projective dimension of $R/I$ when $\mathcal{H} (I)$ is a string or a cycle hypergraph.

Let $(R,\mathfrak {m})$ be a Noetherian local ring which is a quotient of a Gorenstein local ring. Let $M$ be a finitely generated $R$-module. In this paper, we study the structure of the canonical module $K(R\ \mathbb {n}\ M)$ of the idealization $R\ \mathbb {n}\ M$ via the polynomial type introduced by Cuong~\cite {C}. In particular, we give a characterization for $K(R\ \mathbb {n}\ M)$ being Cohen-Macaulay and generalized Cohen-Macaulay.

In this paper, we give complete characterizations of Noetherian domains and integrally closed domains in which every ideal is projectively equivalent to a prime ideal. We also characterize pullbacks satisfying this property and show how to construct integral domains in which every ideal is projectively equivalent to a prime ideal outside the context of Noetherian domains and integrally closed domains.

The notion of Ulrich ideals was introduced by Goto et al.~\cite {GOTWY}. They developed an interesting theory on Ulrich ideals. In particular, they gave a characterization of Ulrich ideals of Gorenstein numerical semigroup rings that are generated by monomials. Using this result, in this paper, we investigate Ulrich ideals of Gorenstein numerical semigroup rings with embedding dimension~3 that are generated by monomials. In particular, we completely determine when such Ulrich ideals are existent in those rings.