2 Rings and Polynomials 30 2.1 Rings, Integral Domains and Fields . . . . . . . . . . . . . . . 30 2.2 Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2.3 Quotient Rings and Homomorphisms . . . . . . . . . . . . . . 33 2.4 The Characteristic of a Ring . . . . . . . . . . . . . . . . . . . 35 2.5 Polynomial Rings . . . . . . . . . . . . . . . . . . . . . . . . . 35 2.6 Ga...

Let f1, . . . , fr ∈ K[x], K a field, be homogeneous polynomials and put F = ∑r i=1 yifi ∈ K[x, y]. The quotient J = K[x, y]/I, where I is the ideal generated by the ∂F/∂xi and ∂F/∂yj , is the Jacobian ring of F . We describe J by computing the cohomology of a certain complex whose top cohomology group is J .

In this paper, we make use of generalized derivations to scrutinize the deportment prime ideal satisfying certain algebraic $*$-identities in rings with involution. specific cases, structure quotient ring $\mathscr{R}/\mathscr{P}$ will be resolved, where $\mathscr{R}$ is an arbitrary and $\mathscr{P}$ a also find behaviour associated involving ideals. Finally, conclude our paper applications pr...

We use quivers and their representations to bring new perspectives on the subregular J-ring JC of a Coxeter system (W,S), subring Lusztig's J-ring. prove that is isomorphic suitable quotient path algebra double quiver (W,S). Up Morita equivalence, such quotients include group algebras all free products finite cyclic groups. then study category mod-AK dimensional right modules AK=K⊗ZJC over an a...

Let G be a finite simple graph on the vertex set $$V(G) = \{x_{1}, \ldots , x_{n}\}$$ and match(G), min-match(G) ind-match(G) matching number, minimum number induced of G, respectively. $$K[V(G)] K[x_{1}, x_{n}]$$ denote polynomial ring over field K $$I(G) \subset K[V(G)]$$ edge ideal G. The relationship between these graph-theoretic invariants ring-theoretic quotient K[V(G)]/I(G) has been stud...

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...

We compare the cohomology ring of flag variety ${\\mathcal{F}\\ell}n$ and Chow Gelfand–Zetlin toric $X{\\operatorname{GZ}}$.We show that $H^(\\mathcal{F}{\\ell}\_n, \\mathbb{Q})$ is Poincaré duality quotient subalgebra $A^(X\_{\\operatorname{GZ}}, generated by degree $1$ elements. compute these algebras for $n=3$ see that, in general, this does not have duality.

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We identify the ring of odd symmetric functions introduced by Ellis and Khovanov as the space of skew polynomials fixed by a natural action of the Hecke algebra at q = −1. This allows us to define graded modules over the Hecke algebra at q = −1 that are ‘odd’ analogs of the cohomology of type A Springer varieties. The graded module associated to the full flag variety corresponds to the quotient...