nLab unique factorization ring



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

Instead of having to constantly distinguish between different notions of unique factorization domains, we could simply generalize to unique factorization rings.


Let RR be a commutative ring. We say that an element rRr\in R is regular if left multiplication by rr is an injection and right multiplication by rr is an injection. We say that an element rRr\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 RR is a unique factorization ring if every regular non-unit has a factorization u=r 1r nu = r_1 \cdots r_n (where n1n \ge 1) as product of irreducibles and this decomposition is unique up to renumbering and rescaling the irreducibles by units.

Put differently: RR is a unique factorization ring precisely when the multiplicative monoid of regular principal ideals of RR (which is isomorphic to the quotient monoid Reg(R)/R ×Reg(R)/R^\times, where Reg(R)Reg(R) denotes the multiplicative subset of regular elements in RR and R ×R^\times denotes the group of units in RR) is a commutative monoid freely generated by irreducible principal ideals.


See also

Created on January 23, 2023 at 17:56:04. See the history of this page for a list of all contributions to it.