Let A be a ring of dimension d containing an infinite field k, T1,…,Tr variables over and P projective A[T1,…,Tr]-module rank n. Assume one the following conditions holds. 2n≥d+3 is extended from A. 2n≥d+2, affine F‾p-algebra singular locus Spec(A) closed set V(J) with ht J≥d−n+2.

A graph G = (V,E) is called a hyper-ring with N nodes (N -HR for short) if V = {0, ..., N − 1} and E = {{u, v}|v − u modulo N is a power of 2}. The following results are shown. We prove that the node-connectivity κ of an N -HR is equal to its degree, say δ, by presenting an algorithm for the explicit construction of δ node-disjoint paths connecting nodes s and t. The length of these paths is bo...

The aim of this paper is to generalize the‎ ‎notion of almost valuation domains to arbitrary commutative‎ ‎rings‎. ‎Also‎, ‎we consider relations between almost valuation rings ‎and pseudo-almost valuation rings‎. ‎We prove that the class of‎ ‎almost valuation rings is properly contained in the class of‎ ‎pseudo-almost valuation rings‎. ‎Among the properties of almost‎ ‎valuation rings‎, ‎we sh...

The answer must be that the radiation is canceled by destructive interference between the radiation fields of the large number N of electrons that make up the steady current. A single electron in uniform circular motion emits electric dipole radiation, whose power is proportional to the square of the acceleration a = v/r, and hence to (v/c). But, the electric dipole moment vanishes for two elec...

Let $R$ be a ring and $M$ a right $R$-module. We call $M$, coretractable relative to $overline{Z}(M)$ (for short, $overline{Z}(M)$-coretractable) provided that, for every proper submodule $N$ of $M$ containing $overline{Z}(M)$, there is a nonzero homomorphism $f:dfrac{M}{N}rightarrow M$. We investigate some conditions under which the two concepts coretractable and $overline{Z}(M)$-coretractable...

Let R be a semiprime ring and let F, f : R → R be (not necessarily additive) maps satisfying F (xy) = F (x)y + xf(y) for all x, y ∈ R. Suppose that there are integers m and n such that F (uv) = mF (u)F (v) + nF (v)F (u) for all u, v in some nonzero ideal I of R. Under some mild assumptions on R, we prove that there exists c ∈ C(I) such that c = (m + n)c2, nc[I, I] = 0 and F (x) = cx for all x ∈...

Let F be an infinite field. For each pair (i, j) with 1 ≤ i, j ≤ n, let Xij : GLn(F ) → F be the coordinate function sending a matrix x to its entry xij . Let F [Xij ] denote the algebra generated by these functions. As this notation suggests, we identify F [Xij ] with the polynomial ring in n 2 indeterminants; this is admissible because the field F is infinite. Let V be a finite-dimensional F ...

Let B be a commutative Noetherian ring, X = Spec B the associated affine scheme, I C B an ideal and V = V{I) C X the closed subset defined by I. DEFINITION. Elements / i , . . . , / a € I define V set-theoretically (equivalent^, V is defined set-theoretically by s equations f\ = 0, f2 = 0 , . . . , fs = 0) (M/i.---./.) = V7. Hubert's Nullstellensatz implies that in the case when B is a finitely...

The mol-ecule of the title compound, C(16)H(16)N(2)O(5)S, has the shape of the letter V but with a small twist; the dihedral angle formed between the benzene rings is 79.66 (9)°. The presence of an intra-molecular N-H⋯O hydrogen bond, leading to an S(6) ring, correlates with the near coplanarity of the carboxyl-ate ester group with the benzene ring to which it is connected. The acetamide residu...