For a standard graded Cohen-Macaulay ring $R$, if the quotient $R/(\underline {x})$ admits non-free totally reflexive modules, where $\underline {x}$ is a system of parameters consisting of elements of degree one, then so does the ring $R$. A non-constructive proof of this statement was given by Takahashi. We will give an explicit construction of the totaly reflexive modules ove $R$ obtained from those over $R/(\underline{x})$.

We consider the question of which Stanley-Reisner rings of graphs admit non-free totally reflexive modules and discuss some examples.

For an Artinian local ring $(R, \mathfrak m)$ with $\mathfrak m^3 =0$ and containing the complex numbers, we describe an explicit construction of uncountably many non-isomorphic indecomposable totally reflexive modules, under the assumption that at least one such non-free module exists.

In this article we observe that $C(X)$ is integral over $I+\mathbb R$ if and only if $I$ is of the form of a finite intersection of real maximal ideals. The integral closure of a subring $I+\mathbb R$ is investigated and it turns out that $I+\mathbb R$ is integrally closed if and only if $I$ is semiprime and $\theta (I)$ is connected. We show that $I^u+\mathbb R$, where $I^u$ is the uniform closure of the ideal $I$, is always a ring of quotients of $I+\mathbb R$ and $C(X)$ is a ring of quotients of $I+\mathbb R$ if and only if $I$ is either essential (large) or a maximal ideal generated by an idempotent. Maximal subrings of the form $I+\mathbb R$ of $C(X)$ are characterized and it is shown that $[M^p]^u+\mathbb R$ (which is isomorphic to $C(X\cup \{p\})$, where $X\cup \{p\}$ is a subspace of $\beta X$) is not contained in a proper maximal subring of $C(X)$. We have observed that for each subring of the form $I+\mathbb R$ of $C(X)$, the sum of every two prime ideals of $I+\mathbb R$ is prime or all of $I+\mathbb R$ if and only if $X$ is an $F$-space. Ideals $I$ are characterized for which $I+\mathbb R$ is a regular ring and for every $z$-ideal $I$, we have shown that the sum of every two $z$-ideals of $I+\mathbb R$ is a $z$-ideal or all of $I+\mathbb R$. Finally, some algebraic and topological properties are given for which $I+\mathbb R$ is a self-injective ring or a uem-ring.

Let $R$ be a commutative, local, Noetherian ring. In a past article, the first author developed a theory of $R$-algebras, termed seeds, that can be mapped to balanced big Cohen-Macaulay $R$-algebras. In prime characteristic $p$, seeds can be characterized based on the existence of certain colon-killers, integral extensions of seeds are seeds, tensor products of seeds are seeds, and the seed property is stable under base change between complete, local domains. As a result, there exist directed systems of big Cohen-Macaulay algebras over complete, local domains. In this work, we will show that these properties can be extended to analogous results in equal characteristic zero. The primary tool for the extension will be the notion of ultraproducts for commutative rings as developed by Schoutens and Aschenbrenner.

We prove the subadditivity property for the maximal degrees of the syzygies of facet ideals of simplicial forests. We do this by interpreting the subadditivity property for any monomial ideal as a property of homologies of its lcm lattice. For an ideal $I$ that is the facet ideal of a simplicial forest, if the $i$-th Betti number is nonzero and $i=a+b$, we show that there are monomials in the lcm lattice of $I$ that are complements in part of the lattice, each supporting a nonvanishing $a$-th and $b$-th Betti number. The subadditivity formula follows from this fact.

The groups of equivalence classes of auto-equivalences of the categories of torsion and torsion-free modules over an integral domain $R$ are determined. For the category of torsion-free modules, this group is isomorphic to the Picard group $\operatorname {Pic} R$ of $R$. We prove that for the category of torsion $R$-modules, this group is isomorphic to a group of certain divisible torsion groups called ``cyclones'' where addition is given in terms of the torsion product. In particular, this group is abelian and contains the direct product of the Picard group of the $R$-completion $\widetilde R$ of $R$ and of the group of ``clones'' of $Q/R$, where $Q$ denotes the quotient field of $R$. The structure of the latter group is described in terms of the inverse system of the unit groups $U(R/Rr)$ of the proper quotients $R/Rr$ ($r \in R$) of $R$.

Let $(R,\mathfrak m)$ be a Noetherian local ring of dimension $d$ and $K,Q$ be $\mathfrak m$-primary ideals in $R$. In this paper we study the finiteness properties of the sets $\Lambda _i^K(R):=\{g_i^K(Q): Q\mbox { is a parameter ideal of }R\}$, where $g_i^K(Q)$ denotes the Hilbert coefficients of $Q$ with respect to $K$, for $1\leq i \leq d$. We prove that $\Lambda _i^K(R)$ is finite for all $1\leq i \leq d$ if and only if $R$ is generalized Cohen-Macaulay. Moreover, we show that if $R$ is unmixed then finiteness of the set $\Lambda _1^K(R)$ suffices to conclude that $R$ is generalized Cohen-Macaulay. We obtain partial results for $R$ to be Buchsbaum in terms of $|\Lambda _i^K(R)|=1$. Our results are more general than previous work of Goto, Ozeki, and others. We also obtain a criterion for the set $\Delta ^K(R):=\{g_1^K(I): I\mbox { is an $\mathfrak m$-primary ideal of }R\}$ to be finite, generalizing a result of Koura and Taniguchi.

We give a negative answer to a question posed by Goodarzi, Pournaki, Seyed Fakhari and Yassemi regarding the $h$-vector of ($S_r$) simplicial complexes.