Cazaran and Kelarev [2] have given necessary and sufficient conditions for an ideal to be the principal; further they described all finite factor rings Zm[X1, · · · , Xn]/I, where I is an ideal generated by an univariate polynomial, which are commutative principal ideal rings. But in [3], Cazaran and Kelarev characterize the certain finite commutative rings as a principal ideal rings. Though, t...

The interpretation of topological resonance energy (TRE), the mathematical measure of energetic chemical aromaticity at the Hückel level of theory, is revisited by providing a concise analysis of the matching polynomial of a chemical graph. Whereas the matching (or acyclic) polynomial is not a properly characteristic polynomial in general, it is the arithmetic mean of the characteristic polynom...

In 2003, Shestakov-Umirbaev solved Nagata’s conjecture on an automorphism of a polynomial ring. In the present paper, we reconstruct their theory by using the “generalized Shestakov-Umirbaev inequality”, which was recently given by the author. As a consequence, we obtain a more precise tameness criterion for polynomial automorphisms. In particular, we deduce that no tame automorphism of a polyn...

A Wedderburn polynomial over a division ring K is a minimal polynomial of an algebraic subset of K. Special cases of such polynomials include, for instance, the minimal polynomials (over the center F = Z(K)) of elements of K that are algebraic over F . In this note, we give a survey on some of our ongoing work on the structure theory of Wedderburn polynomials. Throughout the note, we work in th...

H-local domains arise in a variety of moduleand ideal-theoretic applications. In the first half of the paper, we survey some of these applications, and collect a number of characterizations and examples of h-local domains. In the second half, we show how diverse examples of h-local Prüfer domains arise as overrings of Noetherian domains and polynomial rings in finitely many variables.

We define a closure operation for rings of mixed characteristic and verify that the closure is a ring. We then show that this closure produces a ring with good properties with respect to its Fontaine ring and give an example to show that rings that are not closed in this sense do not satisfy these properties.

We explore the subrings in trigonometric polynomial rings and their factorization properties. Consider the ring S′ of complex trigonometric polynomials over the field Q(i) (see [11]). We construct the subrings S′ 1, S′ 0 of S′ such that S′ 1 ⊆ S′ 0 ⊆ S′. Then S′ 1 is a Euclidean domain, whereas S′ 0 is a Noetherian HFD. We also characterize the irreducible elements of S′ 1, S′ 0 and discuss amo...

We introduce symmetrizing operators of the polynomial ring A[x] in the variable x over a ring A. When A is an algebra over a field k these operators are used to characterize the monic polynomials F (x) of degree n in A[x] such that A ⊗k k[x](x)/(F (x)) is a free A–module of rank n. We use the characterization to determine the Hilbert scheme parameterizing subschemes of length n of k[x](x). Intr...

Let R be an associative ring with 1 and R the multiplicative group of invertible elements of R. In the paper, subgroups of R which may be regarded as analogues of the similitude group of a non-degenerate sesquilinear reflexive form and of the isometry group of such a form are defined in an abstract way. The main result states that a unipotent abstractly defined similitude must belong to the cor...