Factorization of elements in noncommutative rings, I

We extend the classical theory of factorization in noncommutative integral domains to the more general classes of right saturated rings and right cyclically complete rings. Our attention is focused, in particular, on the factorizations of right regular elements into left irreducible elements. We study the connections among such factorizations, right similar elements, cyclically presented modules of Euler characteristic $0$ and their series of submodules. Finally, we consider factorizations as a product of idempotents.