Arbitrary triple systems admitting a multiplicative basis

摘要

Let (T, ) be a triple system of arbitrary dimension, over an arbitrary base field F and in which any identity on the triple product is not supposed. A basis B = {e(i)}(i is an element of l) of T is called niultiPlicative if for any i,j,k is an element of l, we have that is an element of Fe-r for scme r is an element of l. show that if T admits a multiplicative basis, hen it decomposes as the orthogonal direct sum T circle plus(k) J(k) of well-described ideals admitting each one a multiplicative basis. Also, the minimality of T is characterized in terms of the multiplicative basis and it is shown that, under a mild condition, the above direct sum is by the farriily of its minimal ideals.