The reaction of diphenylcyclopropenone with several N-substituted hydroxylamine and primary amines are investigated. Most of these reactions led to ring opened or ring enlarged products, instead of the desired N-Oxide or imine. The reaction of cyclopropeneimine with peracid-another possible route to N-substituted cyclopropeneimine N-Oxide is studied. Most of the unexpected products are ide...

Usually we shall just call A an algebra if the field k is clear from the context. The algebra A is associative if multiplication is associative i.e. for all a, b, c ∈ A, (ab)c = a(bc), and unital if there is a multiplicative identity, i.e. an element usually denoted by 1 such that, for all a ∈ A, 1a = a1 = a. Note that, in this case, 1 = 0 ⇐⇒ A = {0}. Otherwise, the map k → A defined by t 7→ t·...

We find the universal module w(M) for functions (that need not be invariants) of finite matroids, defined on a minor-closed class M and with values in any module L over any commutative and unitary ring, that satisfy a parametrized deletion-contraction identity, F (M) = δeF (M r e) + γeF (M/e), when e is neither a loop nor a coloop. (F is called a (parametrized) weak Tutte function.) Within the ...

Let C be a small category and k a field. There are two interesting mathematical subjects: the category algebra kC and the classifying space |C| = BC. We study the ring homomorphism HH∗(kC) → H∗(|C|, k) and prove it is split surjective, using the factorization category of Quillen [16] and certain techniques from functor cohomology theory. This generalizes the well-known results for finite groups...

KrulΓs principal ideal theorm [Krull] states that q elements in the maximal ideal of a local noetherian ring generate an ideal whose minimal components are all of height at most q. Writing R for the ring, we may consider the q elements, x19 , xq say, as coordinates of an element xeR. It is an easy observation that every homomorphism R —> R carries x to an element of the ideal generated by xi9 ,...

Suppose R and S are local rings and (R,m) → (S,n) is a flat local homomorphism. Given a finitely generated S-module N , we say N is extended (from R) provided there is an R-module M such that S ⊗R M is isomorphic to N as an S-module. If such a module M exists, it is unique up to isomorphism (cf. [EGA, (2.5.8)], and it is necessarily finitely generated. The m-adic completion R → R̂ and the Hensel...

There are many instances of the principle that if A is an algebra of functions on X, then every ring homomorphism A → C is given by evaluation at a particular x ∈ X. Examples are the nullstellensatz in algebraic geometry and the result of Gelfand when X is a compact Hausdorff space. In these cases one can regard X as being included in Hom(A, C) as the set of those f : A → C which satisfy the se...

In this paper, we study the fundamental algorithmic problem of determining the homomorphisms from an Amodule M to an A-module N over an algebra A. We shall outline an algorithm for the case that A is a finite dimensional algebra over a finite field F and both modules M and N are finitely generated as A-modules. More specifically, the input to our algorithm consists of the following data: The mo...

Let $R$ be a unitary ring with an endomorphism $σ$ and $F∪{0}$ be the free monoid generated by $U={u_1,…,u_t}$ with $0$ added, and $M$ be a factor of $F$ setting certain monomial in $U$ to $0$, enough so that, for some natural number $n$, $M^n=0$. In this paper, we give a sufficient condition for a ring $R$ such that the skew monoid ring $R*M$ is quasi-Armendariz (By Hirano a ring $R$ is called...

 Let $R$ be a ring‎, ‎and let $n‎, ‎d$ be non-negative integers‎. ‎A right $R$-module $M$ is called $(n‎, ‎d)$-projective if $Ext^{d+1}_R(M‎, ‎A)=0$ for every $n$-copresented right $R$-module $A$‎. ‎$R$ is called right $n$-cocoherent if every $n$-copresented right $R$-module is $(n+1)$-coprese-nted‎, ‎it is called a right co-$(n,d)$-ring if every right $R$-module is $(n‎, ‎d)$-projective‎. ‎$R$...