Arbitrary algebras with a multiplicative basis

Let A be an algebra of arbitrary dimension, over an arbitrary base field F and in which any identity on the product is not supposed. A basis B={ei}i∈I of A is called multiplicative if for any i,j∈I we have that eiej∈Fek for some k∈I. We show that if A admits a multiplicative basis then it decomposes as the direct sum A=⨁kIk of well-described ideals admitting each one a multiplicative basis. Also the minimality of A is characterized in terms of the multiplicative basis and it is shown that, under a mild condition, the above direct sum is by means of the family of its minimal ideals admitting a multiplicative basis.