In this paper, we give a generalized Cline?s formula for the Drazin inverse. Let R be ring, and let a, b, c, d ? satisfying (ac)2 = (db)(ac), (db)2 (ac)(db), b(ac)a b(db)a, c(ac)d c(db)d. Then ac Rd if only bd Rd. case, (bd)d b((ac)d)2d: We also present formulas group inverses. Some weaker conditions in Banach algebra are investigated. These extend main results of on g-Drazin inverse Liao, Chen...

Using a combinatorial approach which avoids geometry, this paper studies the ring structure of KT (G/B), the T -equivariant K-theory of the (generalized) flag variety G/B. Here, the data G ⊇ B ⊇ T is a complex reductive algebraic group (or symmetrizable Kac-Moody group)G, a Borel subgroup B, and a maximal torus T , and KT (G/B) is the Grothendieck group of T -equivariant coherent sheaves on G/B...

In this paper, we examine the “derived completion” of the representation ring of a pro-p group G∧ p with respect to an augmentation ideal. This completion is no longer a ring: it is a spectrum with the structure of a module spectrum over the Eilenberg–MacLane spectrum HZ , and can have higher homotopy information. In order to explain the origin of some of these higher homotopy classes, we defin...

Hochster established the existence of a commutative noetherian ring R and a universal resolution U of the form 0 ! R ! R ! R ! 0 such that for any commutative noetherian ring S and any resolution V equal to 0! S ! S ! S g ! 0, there exists a unique ring homomorphism R ! S with V = U R S. In the present paper we assume that f = e + g and we nd a resolution of R by free P-modules, where P is a po...

We construct two new G-equivariant rings: K (X, G), called the stringy K-theory of the G-variety X, and H (X, G), called the stringy cohomology of the G-variety X, for any smooth, projective variety X with an action of a finite group G. For a smooth Deligne-Mumford stack X , we also construct a new ring Korb(X ) called the full orbifold K-theory of X . We show that for a global quotient X = [X/...

We prove, under some mild conditions, that the equivariant twisted K-theory group of a crossed module admits a ring structure if the twisting 2-cocycle is 2multiplicative. We also give an explicit construction of the transgression map T1 : H ∗(Γ • ;A) → H((N ⋊ Γ) • ;A) for any crossed module N → Γ and prove that any element in the image is ∞-multiplicative. As a consequence, we prove that, unde...

For any linear algebraic group G, we define a ring CHBG, the ring of characteristic classes with values in the Chow ring (that is, the ring of algebraic cycles modulo rational equivalence) for principal G-bundles over smooth algebraic varieties. We show that this coincides with the Chow ring of any quotient variety (V −S)/G in a suitable range of dimensions, where V is a representation of G and...

Let $R= \oplus_{ \alpha \in G} R_{\alpha}$ be a commutative ring with unity graded by an arbitrary grading monoid $G$. For each positive integer, the notions of graded-n-coherent module and are introduced. In this paper many results generalized from $n$-coherent rings to graded-$n$-coherent rings. last section, we provide necessary sufficient conditions for trivial extension graded-valuation ri...

For m a non-negative integer and G a Coxeter group, we denote by QIm(G) the ring of m-quasiinvariants of G, as defined by Chalykh, Feigin, and Veselov. These form a nested series of rings, with QI0(G) the whole polynomial ring, and the limit QI∞(G) the usual ring of invariants. Remarkably, the ring QIm(G) is freely generated over the ideal generated by the invariants of G without constant term,...

A ring R is called a right Ikeda-Nakayama (for short IN-ring) if the left annihilator of the intersection of any two right ideals is the sum of the left annihilators, that is, if (I ∩ J) = (I) + (J) for all right ideals I and J of R. R is called Armendariz ring if whenever polynomials f (x) = a0 + a1x + ··· + amx, g(x) = b0 + b1x + ··· + bnx ∈ R[x] satisfy f (x)g(x) = 0, then aibj = 0 for each ...