This paper concerns the unique factorisation property in commutative rings not necessarily with identity. We give a new definition of irreducibility and associates in a commutative ring with 1 (crwl), and define a UFR R in terms of a monomorphism from R into a crwl. This becomes equivalent to the definition in  when R has an identity. We generalize results on direct sums and direct summands. By our definition we have new members of the family of UFR's.
AĞARGÜN, A. G. and FLETCHER, C.R. (1997) "Unique Factorisation for Commutative Rings Without Identity," Turkish Journal of Mathematics: Vol. 21: No. 4, Article 2. Available at: https://journals.tubitak.gov.tr/math/vol21/iss4/2