Let $A$ and $B$ be commutative rings with unity, $f:A\to B$ a ring homomorphism and $J$ an ideal of $B$. Then, the subring $A\bowtie ^fJ:=\{(a,f(a)+j)\mid a\in A$ and $j\in J\}$ of $A\times B$ is called the amalgamation of $A$ with $B$ along $J$ with respect to $f$. In this paper, we study the property of Cohen-Macaulay in the sense of ideals, which was introduced by Asgharzadeh and Tousi \cite {AT}, a general notion of the usual Cohen-Macaulay property (in the Noetherian case), on the ring $A\bowtie ^fJ$. Among other things, we obtain a generalization of the well-known result of when Nagata's idealization is Cohen-Macaulay.

Let $T$ be a complete local (Noetherian) equidimensional ring with maximal ideal $\mathfrak{m} $ such that the Krull dimension of $T$ is at least two and the depth of $T$ is at least two. Suppose that no integer of $T$ is a zerodivisor and that $|T|=|T/\mathfrak{m} |$. Let $d$ and $t$ be integers such that $1\leq d \leq \dim T-1$, $0 \leq t \leq \dim T - 1$ and $d - 1 \leq t$. Assume that, for every $\mathfrak{p} \in Ass T$, $ht \mathfrak{p} \leq d-1$ and that if $z$ is a regular element of $T$ and $Q \in Ass (T/zT)$, then $ht Q \leq d$. We construct a local unique factorization domain $A$ such that the completion of $A$ is $T$ and such that the dimension of the formal fiber ring at every height one prime ideal of $A$ is $d - 1$ and the dimension of the formal fiber ring of $A$ at $(0)$ is $t$.

For a commutative ring $R$, one can form a graph $\Gamma (R)^*$ whose vertices are the zero divisors of $R$ (including $0$) and whose edges are the pairs $\{a,b\}$ where $ab=0$ with $a\ne b$. For this graph, a clique is a nonempty subset $X$ such that $ab=0$ for all $a\ne b$ in $X$. If $R$ is a finite ring, there is always a maximum clique of $\Gamma (R)^*$--a clique $X$ such that $|X|\ge |Y|$ for all cliques $Y$. We say that a finite ring $R$ has the clique ideal property if each maximum clique of $\Gamma (R)^*$ is an ideal of $R$. If $R=S\oplus T$ where both $S$ and $T$ are finite rings with the clique ideal property and neither $S$ nor $T$ is a field, then $R$ has the clique ideal property. The converse does not hold. For each positive integer $n>1$, the ring $R=\mathbb Z_n[\scriptstyle{mathrm{X}} ]/(\scriptstyle{mathrm{X}} ^2)$ is a finite ring with the clique ideal property. In contrast, $\mathbb {Z}_n$ has the clique ideal property if and only if $n$ is either a prime or a perfect square.

Let $(R,\frak{m} )$ be a Cohen-Macaulay local ring of positive dimension $d$ and infinite residue field. Let $I$ be an $\frak{m} $-primary ideal of $R$, and let $J$ be a minimal reduction of $I$. In this paper, we show that, if $\widetilde {I^k}=I^k$ and $J\cap I^n=JI^{n-1}$ for all $n\geq k+2$, then $\widetilde {I^n}=I^n$ for all $n\geq k$. As a consequence, we can deduce that, if $r_J(I)=2$, then $\widetilde {I}=I$ if and only if $\widetilde {I^n}=I^n$ for all $n\geq 1$. Moreover, we recover some main results of \cite {Cpv, G}. Finally, we give a counter example for Puthenpurakal [Question 3]{P1}.

We give lower bounds for the degree of the discriminant with respect to $y$ of squarefree polynomials $f\in \mathbb {K}[x,y]$ over an algebraically closed field of characteristic zero. Depending on the invariants involved in the lower bound, we give a geometrical characterization of those polynomials having minimal discriminant, and we give an explicit construction of all such polynomials in many cases. In particular, we show that irreducible monic polynomials with minimal discriminant coincide with coordinate polynomials. We obtain analogous partial results for the case of nonmonic or reducible polynomials by studying their $GL_2(\mathbb {K}[x])$-orbit and by establishing some combinatorial constraints on their Newton polygon. Our results suggest some natural extensions of the embedding line theorem of Abhyankar-Moh and of the Nagata-Coolidge problem to the case of unicuspidal curves of $\mathbb {P}^1\times \mathbb {P}^1$.