Regular ideal

Abstract: The content of a polynomial f over a commutative ring R is the ideal c(f) of R generated by the coefficients of f. If c(fg) = c(f)c(g) for each polynomial g ∈ R[x], then f is said to be Gaussian. If c(f) is an invertible ideal of R, then f is Gaussian. An open question has been whether the converse holds for a polynomial whose content is a regular ideal of R. The main theorem shows slightly more than this; namely, if c(f) has no nonzero annihilators, then c(f) Hom R (c(f), R) = R.

Abstract: An ideal I of a commutative ring R with identity is called a cancellation ideal if whenever IB = IC for ideals B and C of R, then B = C We show that an ideal I is a cancellation ideal if and only if I is locally a regular principal ideal Let R be a commutative ring with identity An ideal I of R is called a cancellation ideal if whenever IB = IC for ideals B and C of R, then B = C It is easily seen that I is a cancellation ideal if and only if whenever IB ⊆ IC for ideals B and C of R, then B ⊆ C A good introduction to cancellation ideals may be found in Gilmer [1, Section 6] As for examples, it is easy to see that a principal ideal (a) is a cancellation ideal if and only if (a) is a regular ideal (ie, a is not a zero divisor) An invertible ideal is a cancellation ideal More generally, an ideal that is locally a regular principal ideal is a cancellation ideal The purpose of this paper is to prove the converse Kaplansky [2, Theorem 287] proved that a finitely generated cancellation ideal in a quasi-local domain is principal We begin with the following lemma which is a modification of Kaplansky’s result (see [1, Exercise 7, page 67]) We use essentially the same argument Lemma Let R be a commutative ring with identity and let I be a cancellation ideal of R Suppose that I = (x, y)+A where A is an ideal of R containing MI for some maximal ideal M Then I = (x) +A or I = (y) +A Proof Put J = ( x + y, xy, xA, yA,A ) Then it is easily checked that IJ = I Since I is a cancellation ideal, we have J = I Thus x = λ ( x + y ) + terms from ( xy, xA, yA,A ) First, suppose that λ ∈ M Since λx ∈ MI ⊆ A, we have x ∈ (y2, xy, xA, yA,A2) LetK = (y)+A Then I = IK Since I is a cancellation ideal, we have I = K Next, suppose that λ / ∈M Then for some μ ∈ R andm ∈M , we have μ (−λ) = 1+m Now −μλy2 = μ (λ− 1)x+ terms from (xy, xA, yA,A2) Since my = (my) y ∈ (MI) y ⊆ Ay, we have y ∈ (x2, xy, xA, yA,A2) Thus, as in the first case, we get that I = (x) +A Theorem Let R be a commutative ring with identity An ideal I of R is a cancellation ideal if and only if I is locally a regular principal ideal Received by the editors May 16, 1996 1991 Mathematics Subject Classification Primary 13A15

Abstract: In 1950 in volume 1 of Proc. Amer. Math. Soc., B. Brown and N. McCoy showed that every (not necessarily commutative) ring $R$ has an ideal $\frak M (R)$ consisting of elements $a$ for which there is an $x$ such that $axa=a$, and maximal with respect to this property. Considering only the case when $R$ is commutative and has an identity element, it is often not easy to determine when $\frak M (R)$ is not just the zero ideal. We determine when this happens in a number of cases: Namely when at least one of $a$ or $1-a$ has a von Neumann inverse, when $R$ is a product of local rings (e.g., when $R$ is $\Bbb Z_{n}$ or $\Bbb Z_{n}[i]$), when $R$ is a polynomial or a power series ring, and when $R$ is the ring of all real-valued continuous functions on a topological space.