Free commutative quasiregular algebras and algebras without quasiregular subalgebras (Q1072600)

Let A be an associative algebra over a field K. If A is a group with respect to the operation \(x\circ y=x+y-x\cdot y,\) then A is called quasiregular. Denote \(A^{\#}=A\dot +K\cdot 1\) (direct sum), which is also quasiregular. The authors give a construction of a free commutative quasiregular algebra in the theorem 1: Let n be a nonzero cardinal, \(A_ n=K(T_ n)\) the quotient field of the algebra of polynomials \(K[T_ n]\). Then the subset \(\hat K_ n=\{h(1-g)^{-1}| h,g\in K[T_ n],\) h, g have the free terms equal \(to\quad 0\}\) is a quasiregular subalgebra of \(A_ n\). This \(\hat K_ n\) is a free quasiregular commutative algebra with \(T_ n\) as the set of free generators. Moreover \(\hat K_ 1\) is a free quasiregular algebra. Another theorem characterizes the quasiregular algebras without quasiregular proper subalgebras. Some examples for showing that there exist algebras which cannot be represented as a direct sum of primitive algebras without quasiregular subalgebras are given in the last section of the paper. See also the authors' paper with the same title in Russ. Math. Surv. 40, No.4, 147-148 (1985); translation from Usp. Mat. Nauk 40, No.4(244), 131- 132 (1985).