Weakly semi-simple finite-dimensional algebras

W. Edwin Clark
1966 Canadian Journal of Mathematics - Journal Canadien de Mathematiques  
Let A be a finite-dimensional (associative) algebra over an arbitrary field F. We shall say that a semi-group 5 is a translate of A if there exist an algebra B over F and an epimorphism : B -» F such that A = 0<£ _1 and S = 1<£ -1 . It is shown in (2) that any such semi-group 5 has a kernel (defined below) that is completely simple in the sense of Rees. Following Stefan Schwarz (4), we define the radical R(S) of S to be the union of all ideals I oi S such that some power I n of / lies in the
more » ... nel K of S. First we prove that the radical of a translate of A is a translate of the radical of A. It follows that A is nilpotent if and only if it has a translate 5 such that R (S) = 5. We then investigate the opposite extreme, i.e., the case in which R(S) = K. If R(S) = K> we shall say that 5 is K-semi-simple. We declare that A is weakly semi-simple if some translate S of A is X-semi-simple. It is shown that A is weakly semi-simple if and only if fAf is semi-simple for some (hence every) principal idempotent / in A ; equivalently, A -fAf ® R(A) (as vector spaces) where R(A) is the radical of A. This result enables us to give a characterization without the use of idempotents of the algebras of class Q studied by R. M. Thrall in (5).
doi:10.4153/cjm-1966-046-7 fatcat:minu6vmbb5br7k7aervv2qgscu