Let $X$ be a Tychonoff space and $A\left(X\right)$ be an intermediate subalgebra of $C\left(X\right)$, i.e., $C^{\ast }\left(X\right)\subseteq A\left(X\right)\subseteq C\left(X\right)$. We show that such subrings are precisely absolutely convex subalgebras of $C\left(X\right)$. An ideal $I$ in $A\left(X\right)$ is said to be a $z_A$-ideal if $Z\left(f\right)\subseteq Z\left(g\right)$, $f\in I$ and $g\in A\left(X\right)$ imply that $g\in I$. We observe that the coincidence of $z_A$-ideals and $z$-ideals of $A\left(X\right)$ is equivalent to the equality $A\left(X\right)=C\left(X\right)$. This shows that every $z$-ideal in $A\left(X\right)$ need not be a $z_A$-ideal and this is a point which is not considered by D. Rudd in Theorem 4.1 of Michigan Math. J. 17 (1970), 139–141, or by G. Mason in Theorem 3.3 and Proposition 3.5 of Canad. Math. Bull. 23:4 (1980), 437-443. We rectify the induced misconceptions by showing that the sum of $z$-ideals in $A\left(X\right)$ is indeed a $z$-ideal in $A\left(X\right)$. Next, by studying the sum of $z$-ideals in subrings of the form $I+ℝ$ of $C\left(X\right)$, where $I$ is an ideal in $C\left(X\right)$, we investigate a wide class of examples of subrings of $C\left(X\right)$ in which the sum of $z$-ideals need not be a $z$-ideal. It is observed that, for every ideal $I$ in $C\left(X\right)$, the sum of any two $z$-ideals in $I+ℝ$ is a $z$-ideal in $I+ℝ$ or all of $I+ℝ$ if and only if $X$ is an $F$-space. This result answers a question raised by Azarpanah, Namdari and Olfati in J. Commut. Algebra 11:4 (2019), 479–509.

Let $\left(Q,\mathfrak{𝔫},k\right)$ be a commutative local Noetherian ring, $f_1,\dots ,f_c$ a $Q$-regular sequence in $\mathfrak{𝔫}$, and $R=Q∕\left(f_1,\dots ,f_c\right)$. Given a complex of finitely generated free $R$-modules, we give a construction of a complex of finitely generated free $Q$-modules having the same homology. A key application is when the original complex is an $R$-free resolution of a finitely generated $R$-module. In this case our construction is a sort of converse to a construction of Eisenbud and Shamash which yields a free resolution of an $R$-module $M$ over $R$ given one over $Q$.

We classify the Hilbert functions of bigraded algebras in $k\left[x_1,x_2,y_1,y_2\right]$ by introducing a numerical function called a Ferrers function.

For a finite-type star operation $\star$ on a domain $R$, we say that $R$ is $\star$-super potent if each maximal $\star$-ideal of $R$ contains a finitely generated ideal $I$ such that (1) $I$ is contained in no other maximal $\star$-ideal of $R$ and (2) $J$ is $\star$-invertible for every finitely generated ideal $J\supseteq I$. Examples of $t$-super potent domains include domains each of whose maximal $t$-ideals is $t$-invertible (e.g., Krull domains). We show that if the domain $R$ is $\star$-super potent for some finite-type star operation $\star$, then $R$ is $t$-super potent, we study $t$-super potency in polynomial rings and pullbacks, and we prove that a domain $R$ is a generalized Krull domain if and only if it is $t$-super potent and has $t$-dimension one.

Let $P$ be a commutative Noetherian ring and $F$ be a self-dual acyclic complex of finitely generated free $P$-modules. Assume that $F$ has length four and $F_0$ has rank one. We prove that $F$ can be given the structure of a differential graded algebra with divided powers; furthermore, the multiplication on $F$ exhibits Poincaré duality. This result is already known if $P$ is a local Gorenstein ring and $F$ is a minimal resolution. The purpose of the present paper is to remove the unnecessary hypotheses that $P$ is local, $P$ is Gorenstein, and $F$ is minimal.

The $b$-vector $\left(b_1,b_2\dots ,b_d\right)$ of a graph $G$ is defined in terms of its clique vector $\left(c_1,c_2\dots ,c_d\right)$ by the equation ${\sum }_{i=1}^d{b}_i{\left(x+1\right)}^{i-1}={\sum }_{i=1}^d{c}_i{x}^{i-1},$ where $d$ is the largest cardinality of a clique in $G$. We study the relation of the $b$-vector of a chordal graph $G$ with some structural properties of $G$. In particular, we show that the $b$-vector encodes different aspects of the connectivity and clique dominance of $G$. Furthermore, we relate the $b$-vector with the Betti numbers of the Stanley–Reisner ring associated to clique simplicial complex of $G$.

The aim of this paper is to give a connection between the $F$-pure threshold of a polynomial and the height of the corresponding Artin–Mazur formal group. For this, we consider a quasihomogeneous polynomial $f\in \mathbb{Z}\left[x_0,\dots ,x_N\right]$ of degree $w$ equal to the degree of $x_0\cdots x_N$ and show that the $F$-pure threshold of the reduction $f_p\in {\mathbb{𝔽}}_p\left[x_0,\dots ,x_N\right]$ is equal to the log-canonical threshold of $f$ if and only if the height of the Artin–Mazur formal group associated to $H^{N-1}\left(X,{\mathbb{𝔾}}_{m,X}\right)$, where $X$ is the hypersurface given by $f$, is equal to 1. We also prove that a similar result holds for Fermat hypersurfaces of degree greater than $N+1$. Furthermore, we give examples of weighted Delsarte surfaces which show that other values of the $F$-pure threshold of a quasihomogeneous polynomial of degree $w$ cannot be characterized by the height.

Let $\mathcal{𝒳}$ and $\mathcal{𝒴}$ be schemes of finite type over $Spec\mathbb{Z}$ and let $\alpha :\mathcal{𝒴}\to \mathcal{𝒳}$ be a finite map. We show the following holds for all sufficiently large primes $p$: If $\varphi$ and $\psi$ are any splittings on $\mathcal{𝒳}\times Spec{\mathbb{𝔽}}_p$ and $\mathcal{𝒴}\times Spec{\mathbb{𝔽}}_p$, such that the restriction of $\alpha$ is compatible with $\varphi$ and $\psi$, and $V$ is any compatibly split subvariety of $\left(\mathcal{𝒳}\times Spec{\mathbb{𝔽}}_p,\varphi \right)$, then the reduction ${\alpha }^{-1}{\left(V\right)}^{red}$ is a compatibly split subvariety of $\left(\mathcal{𝒴}\times Spec{\mathbb{𝔽}}_p,\psi \right)$. This is meant as a tool to aid in listing the compatibly split subvarieties of various classically split varieties.

We study the sets of semistar and star operations on a semilocal Prüfer domain, with an emphasis on which properties of the domain are enough to determine them. In particular, we show that these sets depend chiefly on the properties of the spectrum and of some localizations of the domain; we also show that, if the domain is $h$-local, the number of semistar operations grows as a polynomial in the number of semistar operations of its localizations.