Chain type decomposition in integral domains

Abstract

Let R be a (skew) integral domain. For 0 a R, a is simple if the interval [aR, RI of principal right ideals of R containing aR is not the union of two proper subintervals of [aR, RI. It is shown that each irredundant factorization of an element of R into simple elements is unique up to multiplication by units. All rings considered are (skew) integral domains, that is, rings with unity without proper zero divisors. Let R be a ring and let R* be the monoid of its nonzero elements. If a, b E R* with aR c bR , the set [aR, bR]= {xRjaR xR c bR} is partially ordered by inclusion. If [aR, bR]= [aR, xR] U [xR, R] we say that xR splits [aR, bR]. Let Ch(a, b)= {xRjxR splits [aR, bR]}. Clearly Ch(a, b) is a chain in [aR, bR]. We write Ch(a) for Ch(a, 1), and this is called the chain for a. If Ch(a)= {aR, R}, a is said to be (right) simple; if Ch(a)= [aR, R], a is said to be (right) rigid. As we shall show, both of these concepts are left-right symmetric. In this paper we are interested in the decomposition of elements into simple elements and the uniqueness of such decompositions. We begin with a general statement from which the left-right symmetry of the definitions given above will follow. PROPOSITION 1. Let R be a ring and let a E R*. The posets [aR, R] and [Ra, R] are dually isomorphic; in particular, if a=xx' then the correspondence xR-Rx' is a bijection which reverses order. PROOF. Let a=xx'=yy'. If xRcyR then x=yz for some z; thus zx' =y' and Ry' c Rx'. Reversing the argument we have xRcyR if Ry' c Rx'. Note that xR =yR if and only if Ry'= Rx' which is the case if and only if z is a unit. COROLLARY 2. An element is right rigid if and only if it is left rigid. If a=xx' then from Proposition 1 we see that xR splits [aR, R] if and only if Rx' splits [Ra, R] from which we deduce the following. COROLLARY 3. An element is right simple if and only if it is left simple. Received by the editors July 5, 1972 and, in revised form, September 11, 1972. AMS (MOS) subject classifications (1970). Primary 16A02; Secondary 13G05.