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Von Neumann regularity within a ring of upper triangular matrices is characterized in terms of how such a matrix and an upper triangular generalized inverse can be constructed. In the case of a finite ring, this yields a method for implicitly counting all upper triangular matrices that admit upper triangular generalized inverses. [ABSTRACT FROM AUTHOR]

A characterization of idempotency is given for triangular matrices with entries from an arbitrary ring. This yields an elementary approach to constructing such matrices. It also reveals structure allowing for the explicit enumeration or implicit counting of all idempotent triangular matrices over a given finite ring. [ABSTRACT FROM AUTHOR]

We find bounds for the number of idempotents in a ring of matrices over an arbitrary finite commutative ring. Using these results, we find a bound on the number of idempotents in an arbitrary finite ring, whereby we improve upon the currently known bounds for the number of idempotents. [ABSTRACT FROM AUTHOR]

Let (A, M) be any finite local (nonzero commutative unital) ring. Then the set of A-algebra isomorphism classes represented by (equivalently, consisting of) rings B such that A ⊂ B is a decomposed (minimal ring) extension is in one-to-one correspondence with the collection of sets { I 1 , I 2 } of ideals I 1 and I 2 of A such that M = I 1 ⊕ I 2 (internal direct sum), with any such { I 1 , I 2 } corresponding to the class represented by the ring B = A / I 1 × A / I 2. If A is not a field, then the (finite) set of A-algebra isomorphism classes represented by (equivalently, consisting of) rings B such that A ⊂ B is a ramified (minimal ring) extension has cardinality at least 2; known examples show that this bound cannot be improved. If X denotes an indeterminate over the field F 4 , then R : = F 2 + X F 4 [ X ] / (X 2) is of minimal cardinality in the class of finite local rings A such that A is not a field and the A-algebra isomorphism classes represented by a ramified extension of A constitute fewer than half of the (finitely many) A-algebra isomorphism classes represented by a minimal ring extension of A. [ABSTRACT FROM AUTHOR]

Let A be a finite local (commutative unital) ring. If A is not a field, there exists a positive integer N such that | B | ≤ N for each ring B such that A ⊂ B is an inert (minimal ring) extension. It follows that A is a (finite) field ⇔ the inert extensions of A form infinitely many (equivalently, denumerably many) A-algebra isomorphism classes ⇔ the minimal ring extensions of A form infinitely many (equivalently, denumerably many) A-algebra isomorphism classes ⇔ there exists a minimal ring extension A ⊂ B such that B is a flat A-module ⇔ there exists a minimal ring extension A ⊂ B such that B is a local ring and A ⊂ B is a Galois ring extension. [ABSTRACT FROM AUTHOR]

In this work, we are going to study the elliptic curves over the ring A 3 = ( F 2 d [ X ] ) / ( X 3 ) , precisely we will establish the short exact sequence which is split, and we will define the group extension E a , b ( A 3 ) of E a 0 , b 0 ( F 2 d ) by Ker ( π 3 ) which seems to be beneficial and interesting in ECC. [ABSTRACT FROM AUTHOR]

In this paper we extend the study of total graphs τ(R) to noncommutative finite ringsR. We prove that τ(R) is connected if and only ifRis not local, and we see that in that case τ(R) is always Hamiltonian. We also find an upper bound for the domination number of τ(R) for all finite ringsR. [ABSTRACT FROM PUBLISHER]

Let 𝒫(n, D) be the set of all monic polynomials in ℤ/nℤ[x] of degree D. A least common multiple for 𝒫(n, D) is a monic polynomial L ∈ ℤ/nℤ[x] of minimal degree such that f divides L for all f ∈ 𝒫(n, D). A least common multiple for 𝒫(n, D) always exists, but need not be unique; however, its degree is always unique. In this article, we establish some bounds for the degree of a least common multiple for 𝒫(n, D), present constructions for common multiples in ℤ/nℤ[x], and describe a connection to rings of integer-valued polynomials over matrix rings. [ABSTRACT FROM PUBLISHER]

In this article, we investigate the relationship between the minimum number of proper subgroups of GL(n, q) whose union is the whole GL(n, q) and the maximum number of elements that pairwise generate GL(n, q). We show that the minimum number of proper subrings of M n (q) whose union is the whole M n (q) is exactly the maximum number of elements that pairwise generate M n (q). [ABSTRACT FROM PUBLISHER]

We investigate non-unique factorization of polynomials in pn[x] into irreducibles. As a Noetherian ring whose zero-divisors are contained in the Jacobson radical, pn[x] is atomic. We reduce the question of factoring arbitrary nonzero polynomials into irreducibles to the problem of factoring monic polynomials into monic irreducibles. The multiplicative monoid of monic polynomials of pn[x] is a direct sum of monoids corresponding to irreducible polynomials in p[x], and we show that each of these monoids has infinite elasticity. Moreover, for every m ∈ , there exists in each of these monoids a product of 2 irreducibles that can also be represented as a product of m irreducibles. [ABSTRACT FROM AUTHOR]

The commuting graph of a ring R, denoted by Γ(R), is a graph whose vertices are all noncentral elements of R and two distinct vertices are joint by an edge whenever they commute. It is conjectured that if R is a ring with identity such that Γ(R) ≈ Γ(Mn(F)), for a finite field F and n ≥ 2, then R ≈ Mn(F). Here we prove this conjecture when n = 2. [ABSTRACT FROM AUTHOR]