Let $f:R\to S$ be a ring homomorphism and J be an ideal of S. Then the subring $R\bowtie ^fJ:=\{(r,f(r)+j)\mid r\in R$ and $j\in J\}$ of $R\times S$ is called the amalgamation of R with S along J with respect to f. In this paper, we characterize when $R\bowtie ^fJ$ is a Hilbert ring. As an application, we provide an example of Hilbert ring with maximal ideals of different heights. We also construct non-Noetherian Hilbert rings whose maximal ideals are all finitely generated (unruly Hilbert rings).

Let R be a commutative ring with identity which is not an integral domain. An ideal I of R is called an annihilating ideal if there exists $r\in R- \{0\}$ such that $Ir=(0)$ . The total graph of nonzero annihilating ideals of R is the graph $\Omega (R)$ whose vertices are the nonzero annihilating ideals of R and two distinct vertices $I,J$ are joined if and only if $I+J$ is also an annihilating ideal of R. We study the strong metric dimension of $\Omega (R)$ and evaluate it in several cases.

If $A$ is a subring of a commutative ring $B$ and if $n$ is a positive integer, a number of sufficient conditions are given for “$A[[X]]$ is $n$-root closed in $B[[X]]$” to be equivalent to “$A$ is $n$-root closed in $B$.” In addition, it is shown that if $S$ is a multiplicative submonoid of the positive integers $\mathbb{P}$ which is generated by primes, then there exists a one-dimensional quasilocal integral domain $A$ (resp., a von Neumann regular ring $A$) such that $S=\{n\in \mathbb{P}|A\,\,\text{is}\,n-\text{root}\,\text{closed}\}$ (resp., $S=\{n\in \mathbb{P}\,|\,\,A[[X]]\,\,\text{is}\,n-\text{root}\,\text{closed}\}$).

Let be a representation of the finite group G over the field . If the order |G| of G is relatively prime to the characteristic of or n = 1 or 2, then it is known that the ring of invariants is Cohen-Macaulay. There are examples to show that need not be Cohen-Macaulay when |G| is divisible by the characteristic of . In all such examples is at least 4. In this note we fill the gap between these results and show that rings of invariants in three variables are always Cohen-Macaulay.

Let D be a PID with infinitely many maximal ideals. J. W. Brewer has asked whether some simple ring extension D[α] of D must have nontrivial Picard group. We show that this question has a negative answer.

In this paper we study when an integral domain is a finite intersection of PID or factorial overrings. We show that any Krull domain is the intersection of a PID and a field. We give several sufficient conditions for a Krull domain to be an intersection of two PID or factorial overrings.