Prime-like Elements and Semi-direct Products in Commutative Banach Algebras

We develop results which show that elements in the radical of a commutative Banach algebra are often precluded from having prime-like properties if we avoid certain exceptional situations involving torsion elements. This makes the proof of the Singer-Wermer conjecture conceptually much clearer. It also motivates the definition of an element havingregular powersand allows us to strengthen our previous results concerning necessary conditions for a commutative Banach algebra A to be the semidirect product of some subalgebra together with a specified principal idealsA#, or, equivalently, concerning necessary conditions for there to be an algebraicsplittingof the short exact sequence0→sA#→A→(A/sA#)→0for some given elementsin A. In particular, we show that if A is aradicalBanach algebra andshas regular powers then no such splitting is possible.