A semiorthogonal sum of monocomposition algebras with unity (Q1317619)

A commutative nondegenerate monocomposition algebra \(A\) over a field \(k\), \(\text{char } k\neq 2\), is a commutative nonassociative algebra with a unit 1 and with a nondegenerate bilinear symmetric function \(n(x,y)\) such that \(n(x^ 2, xy)= n(x,x) n(x,y)\). Let \(B\) be the orthocomplement of the unit with respect to \(n(x,y)\). The algebra \(A\) is a semiorthogonal sum of its subalgebras \(C\) and \(D\) if (i) \(B\) is the orthogonal sum of \(C\cap B\) and \(C\cap D\); (ii) \(BC\subseteq k1+B\). Assume that the field \(k\) has no quadratic extensions. Let \(I\) be a proper ideal of \(A\). Then \(I\cap B\) is a proper ideal of \(B\). In particular, if \(A\) is simple so is \(B\). If \(B\) is finite dimensional then \(A\) is simple. The proofs of these results are based on the following Theorem: Let \(A\) be a finite dimensional commutative monocomposition algebra, \(\dim A\geq 3\). If \(A\) has a proper ideal then the ideal is unique and \(\dim A\) is even. The results of the paper generalize the results of the author's paper [Algebra Logic 18, 401--407 (1980); translation from Algebra Logika 18, 637--647 (1979; Zbl 0448.17002)] on simple orthogonal sums of monocomposition algebras.