As defined by Ye [{\bf12}], a ring is semiclean if every element is the sum of a unit and a periodic element. Ahn and Anderson [{\bf1}] called a ring {weakly clean} if every element can be written as $u+e$ or $u-e$, where $u$ is a unit and $e$ an idempotent. A weakly clean ring is {semiclean}. We show the existence of semiclean rings that are not weakly clean. Every semiclean ring is $2$-clean. New classes of semiclean subrings of $\r$ and $\c$ are introduced and conditions are given when these rings are clean. Cleanliness and related properties of $C(X,A)$ are studied when $A$ is a dense semiclean subring of $\r$ or $\c$.

Given a commutative Noetherian local ring, we provide a criterion under which a minimal totally acyclic complex of free modules has symmetric growth. As a special case, we show that, whenever an image in the complex has finite complete intersection dimension, then the complex has symmetric polynomial growth.

In this paper we investigate the class of rigid monomial ideals and characterize them by the fact that their minimal resolution has a unique $\mathbf{Z}^d$-graded basis. Furthermore, we show that certain rigid monomial ideals are lattice-linear, so their minimal resolution can be constructed as a poset resolution. We then give a description of the minimal resolution of a larger class of rigid monomial ideals by appealing to the structure of $\mathcal{L}(n)$, the lattice of all lcm-lattices of monomial ideals on $n$ generators. By fixing a stratum in $\mathcal{L}(n)$ where all ideals have the same total Betti numbers, we show that rigidity is a property which propagates upward in $\mathcal{L}(n)$. This allows the minimal resolution of any rigid ideal contained in a fixed stratum to be constructed by relabeling the resolution of a rigid monomial ideal whose resolution has been constructed by other methods.

Given a power series ring $R^*$ over a Noetherian integral domain $R$ and an intermediate field $L$ between $R$ and the total quotient ring of $R^*$, the integral domain $A = L \cap R^*$ often (but not always) inherits nice properties from $R^*$ such as the Noetherian property. For certain fields $L$ it is possible to approximate $A$ using a localization $B$ of a particular nested union of polynomial rings over $R$ associated to $A$; if $B$ is Noetherian, then $B = A$. If $B$ is not Noetherian, we can sometimes identify the prime ideals of $B$ that are not finitely generated. We have obtained in this way, for each positive integer $m$, a three-dimensional local unique factorization domain $B$ such that the maximal ideal of $B$ is two-generated, $B$ has precisely $m$ prime ideals of height~2, each prime ideal of $B$ of height~2 is not finitely generated and all the other prime ideals of $B$ are finitely generated. We examine the structure of the map $\text{Spec\,} A \to \text{Spec\,} B$ for this example. We also present a generalization of this example to dimension four. This four-dimensional, non-Noetherian local unique factorization domain has exactly one prime ideal $Q$ of height three, and $Q$ is not finitely generated.

Let $A$ be a $\q$-domain, $K=\text{frac\,}(A)$, $B=A^{[n]}$ and $D\in \text{LND}_A(B)$. Assume rank $D=\text{rank\,}D_K=r$, where $D_K$ is the extension of $D$ to $K^{[n]}$. Then we show that

\hskip5pt(i) If $D_K$ is rigid, then $D$ is rigid.

\hskip2.5pt(ii) Assume $n=3$, $r=2$ and $B=A[X,Y,Z]$ with $DX=0$. Then $D$ is triangulable over $A$ if and only if $D$ is triangulable over $A[X]$. In case $A$ is a field, this result is due to Daigle.

We give a constructive proof of Gilmer's theorem that if every nonzero polynomial over a field $k$ has a root in some fixed extension field $E$, then each polynomial in $k[X]$ splits in $E[X]$. Using a slight generalization of this theorem, we construct, in a functorial way, a commutative, discrete, von Neumann regular $k$-algebra $A$ so that each polynomial in $k[X]$ splits in $A[X]$.

Let $V$ be a complex representation of a finite group $G$ of order $g$. Derksen conjectured that the $p$th syzygies of the invariant ring $\text{Sym\,}(V)^G$ are generated in degrees $\le (p+1)g$. We point out that a simple application of the theory of twisted commutative algebras\endash/using an idea due to Weyl\endash/gives the weaker bound $pg^3$, almost for free.

Let $I\subseteq R=\kk[x_1,\ldots,x_n]$ be a homogeneous ideal generated by forms of degree~$r$. We show here that the shapes of the Betti tables of the ideals $I^d$ stabilize, in the sense that there exists some $D$ such that for all $d\geq D$, $\b_{i,j+rd}(I^d)\neq 0\Leftrightarrow \b_{i,j+rD}(I^D)\neq 0$. We also produce upper bounds for the stabilization index $\text{Stab\,}(I)$. This strengthens the result of Cutkosky, Herzog and Trung that the Castelnuovo-Mumford regularity $\text{reg\,}(I^d)$ is eventually a linear function in $d$.