in this paper, we introduce a class of $j$-quasipolar rings. let $r$ be a ring with identity. an element $a$ of a ring $r$ is called {it weakly $j$-quasipolar} if there exists $p^2 = pin comm^2(a)$ such that $a + p$ or $a-p$ are contained in $j(r)$ and the ring $r$ is called {it weakly $j$-quasipolar} if every element of $r$ is weakly $j$-quasipolar. we give many characterizations and investiga...

We propose a suitable substitute for the classical Grothendieck ring of an algebraically closed field, in which any quasi-projective scheme is represented with its non-reduced structure. This yields a more subtle invariant, called the schemic Grothendieck ring. In order to include open subschemes and their complements, we introduce formal motives. Although originally cast in terms of definabili...

For a field F of characteristic different from 2, containing a square root of -1, endowed with an F-compatible valuation v such that the residue field has at most two square classes, we use a combinatorial analogue of the Witt ring of F to prove that an anisotropic quadratic form over F with even dimension d, trivial discriminant and Hasse-Witt invariant can be written in the Witt ring as the s...

We study state space equations within the white noise space setting. A commutative ring of power series in a countable number of variables plays an important role. Transfer functions are rational functions with coefficients in this commutative ring, and are characterized in a number of ways. A major feature in our approach is the observation that key characteristics of a linear, time invariant,...

The space of m-harmonic polynomials related to a Coxeter group G and a multiplicity function m on its root system is defined as the joint kernel of the properly gauged invariant integrals of the corresponding generalised quantum Calogero-Moser problem. The relation between this space and the ring of all quantum integrals of this system (which is isomorphic to the ring of corresponding quasiinva...

The Brauer group of a commutative ring is an important invariant of a commutative ring, a common journeyman to the group of units and the Picard group. Burnside rings of finite groups play an important rôle in representation theory, and their groups of units and Picard groups have been studied extensively. In this short note, we completely determine the Brauer groups of Burnside rings: they van...

We formulate a series of conjectures (and a few theorems) on the quotient of the polynomial ring Q[Z1 xn, y1,.. . , yn] in two sets of variables by the ideal generated by all Sn invariant polynomials without constant term. The theory of the corresponding ring in a single set of variables X = { x 1 , . . . , xn} is classical. Introducing the second set of variables leads to a ring about which li...

in this paper, we investigate various kinds of extensions of twin-good rings. moreover, we prove that every element of an abelian neat ring r is twin-good if and only if r has no factor ring isomorphic to z2  or z3. the main result of [24] states some conditions that any right self-injective ring r is twin-good. we extend this result to any regular baer ring r by proving that every element of a...

Consider the polynomial ring in countably infinitely many variables over a field of characteristic zero, together with its natural action of the infinite general linear group G. We study the algebraic and homological properties of finitely generated modules over this ring that are equipped with a compatible G-action. We define and prove finiteness properties for analogues of Hilbert series, sys...

We study the half-BPS mesonic chiral ring of the N = 1 superconformal quiver theories arising from N D3-branes stacked at Y pq and Labc Calabi-Yau conical singularities. We map each gauge invariant operator represented on the quiver as an irreducible loop adjoint at some node, to an invariant monomial, modulo relations, in the gauged linear sigma model describing the corresponding bulk geometry...