Let F be a field. We shall let W(F) denote the Wit t ring of equivalence classes of nondegenerate symmetric bilinear forms over F. There are many questions one migh t ask about the extent to which one can prescribe the Wit t ring of an extension field of F and the natural homomorphism between these Wit t rings induced by the field inclusion. In this note we consider a formally real field F and ...

For any ring homomorphism R −→ R′ of noetherian rings, the “extension of scalars” morphism from (Sch/R) to (Sch/R′) which sends X/R to X ⊗R R′/R′, extends to a group homomorphism from K0(Sch/R) to K0(Sch/R). Suppose A is an abelian group, and ρ is an “additive function” from (Sch/R) to A, i.e., a rule which assigns to each X ∈ (Sch/R) an element ρ(X) ∈ A, such that ρ(X) depends only on the isom...

Method of secure evaluation of polynomial y = F(x1, ..., xk) over some rings on untrusted computer is proposed. Two models of untrusted computer are considered: passive and active. In passive model untrusted computer correctly computes polynomial F and tries to know secret input (x1, ..., xk) and output y. In active model untrusted computer tries to know input and output and tries to change cor...

Let R be a commutative ring and I an ideal in R which is locally generated by a regular sequence of length d. Then, each f. g. projective R/I-module V has an Rprojective resolution P. of length d. In this paper, we compute the homology of the n-th Koszul complex associated with the homomorphism P1 → P0 for all n ≥ 1, if d = 1. This computation yields a new proof of the classical Adams-Riemann-R...

We show that the kernel I of the ring homomorphism R[yij | i, j ∈ N, i > j]→ R[si, ti | i ∈ N] determined by yij 7→ sisj +titj is generated by two types of polynomials: off-diagonal 3 × 3-minors and pentads. This confirms a conjecture by Drton, Sturmfels, and Sullivant on the Gaussian two-factor model. Our proof is computational: inspired by work of Aschenbrenner and Hillar we introduce the con...

Let R be a commutative ring and I an ideal in R which is locally generated by a regular sequence of length d. Then, each f. g. projective R/I-module V has an Rprojective resolution P. of length d. In this paper, we compute the homology of the n-th Koszul complex associated with the homomorphism P1 → P0 for all n ≥ 1, if d = 1. This computation yields a new proof of the classical Adams-Riemann-R...

In this article, the notion of $n-$derivation is introduced for all integers $ngeq 2$. Although all derivations are $n-$derivations,  in general these notions are not equivalent. Some properties of ordinary derivations are  investigated for $n-$derivations. Also, we show that under certain mild condition  $n-$derivations are derivations.

In this paper we study the notion of Smarandache loops. We obtain some interesting results about them. The notion of Smarandache semigroups homomorphism is studied as well in this paper. Using the definition of homomorphism of Smarandache semigroups we give the classical theorem of Cayley for Smarandache semigroups. We also analyze the Smarandache loop homomorphism. We pose the problem of findi...