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This paper studies n -clean properties in the setting of principal ideal domains through a case study involving subrings of Dubrovin valuation rings inside matrix rings. Let S = Z ( 2 ) = m n | m , n ∈ Z , gcd ( 2 , n ) = 1 be the localization of the principal ideal domain Z at the prime ideal (2), and consider subring R = [ a 2 b 2 c d ] | a , b , c , d ∈ Z ( 2 ) of the simple Artinian ring Q = M 2 ( Q ) . We show that R forms a Dubrovin valuation order in Q and investigate its clean-type structure. Moreover, we prove that R is clean and strongly clean, and consequently n -clean for all positive integers n . Furthermore, we construct an explicit example of a Dubrovin valuation ring that is not clean, demonstrating that the Dubrovin valuation condition does not generally imply cleanness. This study clarifies how n -clean decompositions interact with valuation-theoretic structures in matrix rings arising from principal ideal domains.

This paper explains that a symmetrical element on a ring having a Moore Penrose general inverse is commutative with a Generalized Moore Penrose inverse. This is due to the properties of the involution in the ring. By using these results and the main properties of Generalized Moore Penrose invers obtained directly from definition Generalized Moore Penrose invers, several theorems that discuss the necessary and sufficient of an element in a ring that has Generalized Moore Penrose invers and also symmetric had been found.

For factorizing representative (or rational) series, with coefficients in a commutative ring A containing ℚ, we examine various products such as concatenation, shuffle and its ϕ -deformations, … (and their co-products) defined on the free monoid which are such that their associated bialgebras are isomorphic to the Sweedler’s dual, for A being a field K .

In this article, an element w of an associative ring R is called m-regular nil clean or m-rnc if expressed as w = a m + b where a m is m-regular element and b is a nilpotent element. R is named m-regular nil clean ring or m-rnc ring. If all the elements of a ring R are m-rnc, some characteristics and basic properties of m-rnc rings are presented in this work.

In this study, we regard \\(M\\) as a semiprime \\(\\)-semiring and introduce the notion of orthogonal \\(( )\\)-derivations within such structures. We explore various characterizations of semiprime \\(\\)-semirings and determine the conditions under which two \\(( )\\)-derivations are orthogonal.

We use the geometry of the stellahedral toric variety to study matroids. We identify the valuative group of matroids with the cohomology ring of the stellahedral toric variety and show that valuative, homological and numerical equivalence relations for matroids coincide. We establish a new log-concavity result for the Tutte polynomial of a matroid, answering a question of Wagner and Shapiro–Smirnov–Vaintrob on Postnikov–Shapiro algebras, and calculate the Chern–Schwartz–MacPherson classes of matroid Schubert cells. The central construction is the ‘augmented tautological classes of matroids’, modeled after certain toric vector bundles on the stellahedral toric variety.

We study noncommutative rings whose proper subrings all satisfy the same chain condition. We show that if every proper subring of a ring \\(R\\) is right Noetherian, then \\(R\\) is either right Noetherian or the trivial extension of \\(Z\\) by the Prüfer \\(p\\)-group for a prime \\(p\\). We also prove that if every proper subring of \\(R\\) is right Artinian, then \\(R\\) is either right Artinian or \\(Z\\). For commutative rings, both results were proved by Gilmer and Heinzer in 1992. Our result for right Artinian subrings only generalises the absolute case of their commutative result. We generalise the full result (when only certain subrings are right Artinian) in the context of PI rings.

We construct a family of semiprimitive and non von Neumann regular rings satisfying that any right or left module is isomorphic to a quotient of its flat cover (in the sense of Enochs) by a small submodule. This answers in the negative a question posed by A.~Amini, B.~Amini, M.~Ershad and H.~Sharif (2007).

For any odd integer$d$, we give a presentation for the integral Chow ring of the stack$\\mathcal {M}_{0}(\\mathbb {P}^r, d)$, as a quotient of the polynomial ring$\\mathbb {Z}[c_1,c_2]$. We describe an efficient set of generators for the ideal of relations, and compute them in generating series form. The paper concludes with explicit computations of some examples for low values of$d$and$r$, and a conjecture for a minimal set of generators.