To Chang Theorem. III

Various multilinear polynomials of Capelli type belonging to a free associative algebra F{X boolean OR Y} over an arbitrary field F generated by a countable set X boolean OR Y are considered. The formulas expressing coefficients of polynomial Chang R(<(x))over bar>,<(y))over bar>vertical bar<(w))over bar>) are found. It is proved that if the characteristic of field I.' is not equal two then polynomial R(<(x))over bar>,<(y))over bar>vertical bar<(w))over bar>) may be represented by different ways in the form of sum of two consequences of standard polynomial S- (<(x))over bar>). The decomposition of Chang polynomial .H(<(x))over bar>,<(y))over bar>vertical bar<(w))over bar>) different from already known is given. Besides, the connection between polynomials R(<(x))over bar>,<(y))over bar>vertical bar<(w))over bar>)and H(<(x))over bar>,<(y))over bar>vertical bar<(w))over bar>) is found. Some consequences of standard polynomial being of great interest for algebras with polynomial identities are obtained. In particular, a new identity of minimal degree for odd component of Z(2)- -graded matrix algebra M-(m,M-m) (F) is given.