Narrow Results

In this paper, we study zero divisors in the Hurwitz series rings and the Hurwitz polynomial rings over general non-commutative rings. We first construct Armendariz rings that are not Armendariz of the Hurwitz series type and find various properties of (the Hurwitz series) Armendariz rings. We show that for a semiprime Armendariz of the Hurwitz series type (so reduced) ring $R$ with $a.c.c.$ on annihilator ideals, HR (the Hurwitz series ring with coefficients over $R$) has finitely many minimal prime ideals, say $B_1,\dots ,B_m$ such that $B_1\cdot \dots \cdot B_m=0$ and $B_i=H{A}_i$ for some minimal prime ideal $A_i$ of $R$ for all $i$, where $A_1,\dots ,A_m$ are all minimal prime ideals of $R$. Additionally, we construct various types of (the Hurwitz series) Armendariz rings and demonstrate that the polynomial ring extension preserves the Armendarizness of the Hurwitz series as the Armendarizness.

Let ${𝒜}={(A_n)}_{n\ge 0}$ be an increasing sequence of rings and $A={\bigcup }_{n\ge 0}{A}_n$. It is shown that the ring $H{𝒜}$ (respectively, $h{𝒜}$) is Noetherian if and only if (i) the ring $A_0$ is Noetherian, (ii) the sequence ${𝒜}$ is stationary, (iii) for each $n\ge 1$, the $A_0$-module $A_n$ is finitely generated and (iv) $\mathbb{Q}\subseteq A_0$. Also, we show that if $A\subseteq B$ is a ring extension and $I$ an ideal of $B$, the following assertions hold: (1) if the ring $H(A,I)$ (respectively, $h(A,I)$) is Noetherian, then (i) the ring $A$ is Noetherian, (ii) the $A$-module $I$ is finitely generated, (iii) the ideal $I$ of $B$ is idempotent and (iv) $\mathrm{\text{charact}}(A)=0$. Conversely, (2) if (i) the ring $A$ is Noetherian, (ii) the $A$-module $I$ is finitely generated, (iii) the ideal $I$ of $B$ is idempotent and (iv) $\mathbb{Q}\subseteq A$, then the ring $H(A,I)$ (respectively, $h(A,I)$) is Noetherian.

A differential quasifield is a natural generalization of a differential field in characteristic p > 0. Elementary properties of differential quasifields are considered, and a generalized version of the theorem on the connection between linear independence over constants and the wronskian is presented. The notion of a partitioned differential quasifield is introduced, and the behavior of partitioned differential quasifields is considered, including under an extension of scalars.

A ring R with Jacobson radical J(R) is a homogeneous semilocal ring if R/J(R) is simple artinian. In this paper, we study the transfer of the property of being homogeneous semilocal from a ring R to the formal power series ring R[[x]], the skew formal power series ring R[[x, α]] and the Hurwitz series ring HR. The results of the paper generalize those proved for commutative local rings. We also consider finite centralizing extensions proving that if the ring of matrices M_n(R) is a homogeneous semilocal ring, then so is R. More generally, if e is an idempotent of a homogeneous semilocal ring S, then eSe is homogeneous semilocal.

Let $R[[X]]$ be the collection of power series with coefficients in a commutative ring $R$ with identity. For a suitable function $\lambda$ from ${\mathbb{N}}_0$ to $\mathbb{N}$, one can define a multiplication ${\ast }_{\lambda }$ in $R[[X]]$ such that together with the usual addition, $R[[X]]$ becomes a ring that contains $R$ as a subring. Denote this ring by $R^{\lambda }[[X]]$. By this observation, the usual power series ring and the well-known Hurwitz series ring are the special cases of $R^{\lambda }[[X]]$ when $\lambda (i)=1$ for all $i$ and $\lambda (i)=i!$ for all $i$, respectively. In this paper, we study the Krull dimension of $R^{\lambda }[[X]]$. We first introduce the concept of almost strong finite type (ASFT) rings and study basic properties of this type of rings. We then show that for any function $\lambda$, the Krull dimension of $R^{\lambda }[[X]]$ is at least $2^{{\aleph }_1}$ if $R$ is a non-ASFT ring, which is an analogue of the result about the Krull dimension of the usual power series ring that $dimR[[X]]\ge 2^{{\aleph }_1}$ if $R$ is a non-SFT ring. In particular, we have the Krull dimension of the Hurwitz series ring is at least $2^{{\aleph }_1}$ if $R$ is a non-ASFT ring, which gives an answer to a question of Benhissi and Koja about the infiniteness of the Krull dimension of the Hurwitz series ring.

In this paper we study some properties of the Hurwitz series ring HA over a commutative ring A, such as the nilradical of HA and the chain condition on its annihilators. We provide an example showing that the last property does not pass from A to HA. A strongly Hopfian ring is a ring satisfying the chain condition on some type of annihilators. We give a large class of strongly Hopfian rings A such that HA are not strongly Hopfian.

Let $R$ and $S$ be two commutative rings with unity, let $J$ be an ideal of $S$ and $\phi :R\to S$ be a ring homomorphism. In this paper, we give a characterization for the amalgamated algebra $R{\bowtie }^{\phi }J$ to be a Nagata ring, a strong S-domain, and a catenarian. Also, we investigate the conditions that the ring of Hurwitz series over $R{\bowtie }^{\phi }J$ has a complete comaximal factorization.