On identities of right alternative metabelian Grassmann algebras

The right alternative metabelian (solvable of index 2) Grassmann algebras of rank 1 and 2 are studied. A basis of identities of a right alternative metabelian Grassmann algebra of rank 1 is presented. Then it is proved that the variety generated by the indicated algebra has almost finite topological rank. It is also shown that the variety generated by a Grassmann algebra of rank 2 is not Spechtian. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 13, No. 2, pp. 157–183, 2007.     

On the completion of excellent rings in characteristicp>0

Si considera il seguente problema posto da Grothendieck (E.G.A.): SeA è un anello eccellente edm un ideale diA, (A, m) ^{^}=m-adico completamento diA è eccellente? Si mostra che la risposta è positiva nei seguenti casi:

1. A=algebra di tipo finito su un DVR completo di caratteristicap>0;
2. A=algebra di tipo finito su un DVRC contenente un corpok di caratteristicap>0 e finito suk [C ^p ] oppure tale che:

1. per ogni sottocampok′ dik contenentek ^p tale che [k:k′]<∞, il modulo universale finito dei differenzialiD _{k′ (C)} esiste;
2. il corpo residuoK diC soddisfa rank _KK ? _K/k <∞
3. C ha una Der-base.

     

On the identities of subalgebras of matrices over the Grassmann algebra

The polynomial identities of certain subalgebras of matrices, over the Grassmann algebra, are studied in terms of their cocharacters. Our present knowledge of such characters for matrices over a fieldF (charF=0) plays a role here, and some of these results are extended to these subalgebras. In particular, we obtain bounds for the codimensions of these algebras (Theorem 0.1 below). Partially supported by an N. S. F. Grant.     

On identities of finite-dimensional nilpotent algebras

Translated from Algebra i Logika, Vol. 30, No. 5, pp. 540–556, September–October, 1991.     

Grassmann manifolds of Jordan algebras

We show that, in a JB-algebra, the projections form a Banach manifold and also, the rank-n projections in a JBW-factor form a Riemannian symmetric space of compact type, for . Received: 18 July 2005     

On identities for coalgebras and hopf algebras

The notion of a polynomial identity for algebras is well-known. In this work the notion of identity is transferred to coalgebras (where one may call it a coidentity). The author studies the basic properties of coidenti-ties, some examples of Hopf algebras with an identity or a coidentity are considered.     

Polynomial identities for matrices over the Grassmann algebra

We determine minimal Cayley–Hamilton and Capelli identities for matrices over a Grassmann algebra of finite rank. For minimal standard identities, we give lower and upper bounds on the degree. These results improve on upper bounds given by L. Márki, J. Meyer, J. Szigeti and L. van Wyk in a recent paper.     

On the identities in algebras generated by linearly connected idempotents

We investigate the problem of the existence of polynomial identities (PI) in algebras generated by idempotents whose linear combination is equal to identity. In the case where the number of idempotents is greater than or equal to five, we prove that these algebras are not PI-algebras. In the case of four idempotents, in order that an algebra be a PI-algebra, it is necessary and sufficient that the sum of the coefficients of the linear combination be equal to two. In this case, these algebras are F₄-algebras.Translated from Ukrainskyi Matematychnyi Zhurnal, Vol. 56, No. 6, pp. 782–795, June, 2004.     

On the existence of approximate identities in ideals of group algebras

A short proof is given of the theorem by Wik, who, in 1965, showed that the nowhere-dense strong Ditkin subsets ofT, the circle group, are finite.     

Shape identities in bernstein algebras

We study the shape identities arising in the theory of Bernstein algebras. We determine all shape identities of minimal degree for two subclasses of Bernstein algebras, namely, normal Bernstein algebras and exceptional Bernstein algebras.