A unique factorization ring is like a unique factorization domain but where we do not require the commutative ring to be an integral domain: instead of talking about the non-zero elements, we simply talk about the regular elements of the commutative ring, since the non-regular elements of a commutative ring are precisely the zero divisors of a commutative ring.
This becomes important in constructive mathematics when we have multiple different notions of integral domain, where one could talk about
the non-regular elements being zero,
or in rings with a tight apartness relation, the elements apart from zero being regular,
or in rings with decidable tight apartness, where both notions coincide
Instead of having to constantly distinguish between different notions of unique factorization domains, we could simply generalize to unique factorization rings.
Let $R$ be a commutative ring. We say that an element $r\in R$ is regular if left multiplication by $r$ is an injection and right multiplication by $r$ is an injection. We say that an element $r\in R$ is a unit if it is invertible. A non-unit is called irreducible if it can not be represented as a product of two non-units.
A commutative ring $R$ is a unique factorization ring if every regular non-unit has a factorization $u = r_1 \cdots r_n$ (where $n \ge 1$) as product of irreducibles and this decomposition is unique up to renumbering and rescaling the irreducibles by units.
Put differently: $R$ is a unique factorization ring precisely when the multiplicative monoid of regular principal ideals of $R$ (which is isomorphic to the quotient monoid $Reg(R)/R^\times$, where $Reg(R)$ denotes the multiplicative subset of regular elements in $R$ and $R^\times$ denotes the group of units in $R$) is a commutative monoid freely generated by irreducible principal ideals.
A unique factorization domain is precisely a unique factorization ring which is an integral domain; i.e. a commutative ring whose multiplicative subset of regular elements are precisely the non-zero elements.
A prefield ring is a unique factorization ring where every regular element is invertible.
Last revised on June 26, 2024 at 16:17:23. See the history of this page for a list of all contributions to it.