Reductive and semisimple algebraic monoids

Let Mbe an irreducible linear algebraic monoid over an algebraically closed field. M is reductive if its unit group G is a reductive (algebraic) group ; M is semisimple if M not equal G and G is reductive with a one dimensiona l center. The following theorems are proved: (1) M is reductive iff M is regular with its kernel (in the sense of semigr oup) a reductive group. (2) M is semisimple iff M is regular with exactly two central idempotents a nd with its kernel, ker(M), a semisimple (algebraic) group iff M-e, the irr educible component of the identity in {a epsilon M \ ae = ea = e}, is a sem isimple monoid and ker(M) is a semisimple group, where e is a minimal idemp otent of M. (3) If dim M = 4 and M not equal G is nonsolvable, then M is semisimple. (4) If dim M = 5 and M not equal G is nonsolvable, then M is reductive iff rank(G) not equal 2. Five dimensional reductive monoids with zero are further analyzed in terms of semisimple monoids.