Schreier varieties of algebras
A variety \({\mathcal X}\) is called a Schreier one if a subalgebra of an \({\mathcal X}\)-free algebra is free. In the sequel the word ``variety'' means ``variety of linear algebras over a field''. Recall that a variety \({\mathcal X}\) is called homogeneous if an identity \(f=0\) implies in \({\mathcal X}\) an identity \(g=0\) whenever \(g\) is a homogeneous component of the polynomial \(f\). If \({\mathcal X}\) is a variety and \(A\in {\mathcal X}\) then \(U_{\mathcal X} (A)\) denotes the universal multiplicative enveloping algebra of \(A\) in \({\mathcal X}\). Let \(F\) be the \({\mathcal X}\)-free algebra over the alphabet \(\{x_1, \dots, x_k, \dots\}\), \(I_F\) the free right \(U_{\mathcal X} (A)\)-module with a base \(\{y_1, \dots, y_k, \dots\}\), \(D: F\mapsto I_F\) the derivation such that \(D(x_i)= y_i\) for all \(i\geq 1\). If \(H\) is a subalgebra of \(F\) then \(J_H\) denotes the submodule of \(I_F\) generated by the set \(\{D(h)\mid h\in H\}\). The main results of the paper are the following three theorems. Theorem 1. Let \({\mathcal X}\) be a homogeneous variety of linear algebras over a field \(F\). Then: 1) if \({\mathcal X}\) is a Schreier variety then the following holds: a) if \(F\) is an \({\mathcal X}\)-free algebra then \(U_{\mathcal X} (F)\) is a free associative algebra; b) if \(F\) is an \({\mathcal X}\)-free algebra and \(H\) is a homogeneous subalgebra of \(F\) then \(U_{\mathcal X} (F)\) is a free right \(U_{\mathcal X} (H)\)-module with a homogeneous base; 2) if \({\mathcal X}\) satisfies the conditions a) and b) and \(\text{char} (F)=0\) then \({\mathcal X}\) is a Schreier variety. Theorem 2. A homogeneous variety \({\mathcal X}\) of linear algebras over a field is a Schreier one whenever it satisfies the condition a) of Theorem 1 and the following condition: c) if \(F\) is an \({\mathcal X}\)-free algebra, \(H\) is a subalgebra of \(F\), \(f\in F\), and \(D(f)\in J_H\) then \(f\in H\). Theorem 3. The variety of linear algebras over a field given by the identity \(xx^2=0\) is a Schreier variety.
universal multiplicative enveloping algebra
scientific article; zbMATH DE number 767663
scientific article