The polynomial ring over a Goldie ring need not be a Goldie ring

Abstract

Rings satisfying chain conditions have been of interest for quite some time. The famous Wedderburn Theorem states that semisimple Artinian rings must be finite direct sums of matrix rings. Goldie’s theorem links rings with ascending chain conditions to rings with descending chain conditions, thus extending Wedderburn’s result. A ring R is said to be a (right) Goldie ring if R satisfies the ascending chain condition on (right) annihilator ideals and R contains no infinite direct sum of (right) ideals. From Goldie’s theorem we know that a semiprime (right) Goldie ring must be an order in a semisimple (right) Artinian ring. A natural question to arise is whether the matrix and polynomial rings over (right) Goldie rings are also necessarily (right) Goldie rings, Under certain additional hypotheses such as if the ring is an order in a (right) Artinian ring [4] or if the ring contains a certain type of uncountable set in its center [I], the answer is yes. Moreover, the second Goldie condition is always preserved [3]. Hence, as in the case for the matrix ring counterexample [2], we must explore the ascending chain condition on (right) annihilator ideals. In the next section we construct a commutative Goldie ring R whose polynomial ring R[t] contains two infinite sets of polynomials, {p,(t)} and (qj(t)}, such that pi(t) qj(t) = 0 iff i #j. This condition forces qk(t) to be in Ann((p,(t): i>k)) and qk(t) to be excluded from Ann({p,(t): i>k1)). Thus R[t] has an infinite ascending chain of annihilator ideals