Given a finite ring A which is a free left module over a subring R of A, two types of R-bases, pseudo-self-dual bases (similar to trace orthogonal bases) and symmetric bases, are defined which in turn are used to define duality preserving maps from codes over A to codes over R. Both types of bases are generalizations of similar concepts for fields. Many illustrative examples are given to shed l...

If every subring of an integral domain is atomic, we say that the latter hereditarily atomic. In this paper, study atomic domains. First, characterize when certain direct limits Dedekind domains are in terms overrings. Then use characterization to determine fields On other hand, investigate hereditary atomicity context rings polynomials and Laurent polynomials, characterizing whose respectively...

The role of symmetry in ring theory is universally recognized. most directly definable universal relation a symmetric set isomorphism. This article develops certain structure bipolar fuzzy subrings, including quotient ring, homomorphism, and We define (?,?)-cut investigate the algebraic attributions this phenomenon. also support prove various important properties relating to concept. Additional...

We study subrings in the Chow ring CH∗(A)Q of an abelian variety A, stable under the Fourier transform with respect to an arbitrary polarization. We prove that by taking Pontryagin products of classes of dimension ≤ 1 one gets such a subring. We also show how to construct finite-dimensional Fourier-stable subrings in CH∗(A)Q. Another result concerns the relation between the Pontryagin product a...

and we call A the zero ring denoted by 0. A ring homomorphism is a mapping f of a ring A into a ring B such that for all x, y ∈ A, f(x + y) = f(x) + f(y), f(xy) = f(x)f(y) and f(1) = 1. The usual properties of ring homomorphisms can be proven from these facts. A subset S of A is a subring of A if S is closed under addition and multiplication and contains the identity element of A. The identity ...

Let IA be the toric ideal of a homogeneous normal configuration A ⊂ Z . We prove that IA is generated by circuits if and only if each unbalanced circuit of IA has a connector which is a linear combination of circuits with a square-free term. In particular, if each circuit of IA with non-square-free terms is balanced, then IA is generated by circuits. As a consequence we prove that the toric ide...

We show how tropical varieties of ideals I E K[x] over a field K with non-trivial valuation can always be traced back to tropical varieties of ideals π−1I E RJtK[x] over some dense subring R in its ring of integers. Moreover, for homogeneous ideals, we present algorithms on how the latter can be computed in finite time, provided that π−1I is generated by elements in R[t, x]. While doing so, we ...

Consider the universal gate set for quantum computing consisting of gates X, CX, CCX, omega^dagger H, and S. All these have matrix entries in ring Z[1/2,i], smallest subring complex numbers containing 1/2 i. Amy, Glaudell, Ross proved converse, i.e., any unitary with Z[1/2,i] can be realized by a circuit over above using at most one ancilla. In this paper, we give finite presentation generators...

An algorithm is constructed that decides if a given finite system of equations over a free Q-group has a solution, and if it does, finds a solution. 0. Introduction Systems of equations over a group have been widely studied (see, for instance, [4],[5],[11]). This is currently one of the main streams of combinatorial group theory. The problem of deciding if a system of equations in a group has a...