Limit T-spaces.
Let \(A\) be a free associative algebra over a field \(F\) with countably many generators without an identity element. Let \(T_V\) be the T-ideal of some variety \(V\) of associative algebras. So \(T_V\) consists of all identities holding in any algebra of \(V\). A subspace \(U\) of the free algebra \(A/T_V\) of the variety \(V\) is called a T-space if for any algebra endomorphism \(\varphi\colon A/T_V\to A/T_V\) we have \(\varphi(U)\subset U\). \(U\) is called a limit T-space if it is not finitely generated as a T-space and any proper T-subspace of \(U\) is finitely generated as a T-space. Now assume \(F\) is of prime characteristic \(p\) and \(V_p\) is the variety of associative algebras without unity defined by the identities \([[x,y],z]=0\) and \(x^4=0\) if \(p=2\) and by the identities \([[x,y],z]=0\) and \(x^p=0\) if \(p>2\) (here \([x,y]=xy-yx\)). Let \(A/T_{V_p}\) be the free algebra of countable rank of the variety \(V_p\) and let \(S\) be the T-space in \(A/T_{V_p}\) generated by \(x_1^2x_2^2\cdots x_k^2+T_{V_2}\), where \(k\in\mathbb{N}\) if \(p=2\), and by \(x_1^{\alpha_1}x_2^{\alpha_2}[x_1,x_2]\cdots x_{2k-1}^{\alpha_{2k-1}}x_{2k}^{\alpha_{2k}}[x_{2k-1},x_{2k}]+T_{V_p}\), where \(k\in\mathbb{N}\) and \(\alpha_1,\dots,\alpha_{2k}\in\{0,p-1\}\) if \(p>2\). The author shows that \(S\) is a limit T-space. As a corollary the author constructs a limit T-space in \(A\).
scientific article; zbMATH DE number 5358200
scientific article