By an essential product of two
rings is meant a subdirect product which contains an essential right ideal of the
direct product. The aim of this paper is to investigate the utility of this concept in
the study of nonsingular rings. The first section derives some basic properties of
essential products and develops some criteria for recognizing essential products. In
the second section, a study of the socles of nonsingular modules leads to a theorem
that any nonsingular ring is an essential product of a ring with essential socle and a
ring with zero socle. The third section is devoted to a theorem which tells when an
essential product can be a splitting ring, i.e., a ring such that the singular
submodule of any right module is a direct summand. In the final section, this
theorem is used to construct two examples of splitting rings of types previously
unknown.
