Defining the Product of Ideals

Solution 1:

Suppose $I=J=(x,y)\subseteq \mathbb R[x,y]$. Then $x^2$ and $y^2$ are in your naive definition, but their sum is not.

It is an enlightening exercise to try to see what exactly $\mathbb Z$ and $\mathbb C[x]$ have that make the naive definition work...

Solution 2:

I think you'll find of interest my old sci.math post below.

From: Bill Dubuque Date: 30 Jul 2003 23:54:07 -0400 Message-ID:

Bill Dubuque wrote:

Rasmus Villemoes wrote: >

In my algebra textbook, the product of two ideals I,J is defined as

{ sum_{i=1..n} a_i b_i | n >= 1 , a_i in I and b_i in J }

Now it is rather easy to prove that IJ is an ideal in R. The last question of the exercise is:

Is A = { ab | a in I, b in J } an ideal of R.

Now the preceding questions strongly suggest that the answer in general is no, but I can't find a counterexample. Clearly, (since it is understood that R is commutative), if one of I or J is a principal ideal, the set A is an ideal, so a counterexample has to consist of a non-PID and two ideals generated by at least two elements each [...]

HINT: Find proper ideals whose product contains an irreducible element,

e.g. p in (p,a)(p,b) if (a,b) = (1)

Examples abound.

Domains where ideals multiply simply as IJ = { ij : i in I, j in J }, are called condensed domains. Below are reviews of related papers.

84a:13019 13F99 Anderson, David F.; Dobbs, David E.
On the product of ideals.
Canad. Math. Bull. 26 (1983), no. 1, 106-114.

In this paper the authors define an integral domain R to be a condensed domain provided IJ = {ij: i in I, j in J} for all ideals I and J of R. Bezout domains are condensed domains. The main results of the paper characterize condensed domains within some large class of domains. For example, it is shown that a GCD-domain is condensed if and only if it is a Bezout domain, and a Krull domain is condensed if and only if it is a principal ideal domain. For a Noetherian domain R to be condensed it is necessary that dim R <= 1 and that the integral closure of R be a principal ideal. Reviewed by J. T. Arnold

86h:13017 13F05 (13B20 13G05)
Anderson, David F.(1-TN); Arnold, Jimmy T.(1-VAPI); Dobbs, David E.(1-TN)
Integrally closed condensed domains are Bezout.
Canad. Math. Bull. 28 (1985), no. 1, 98-102.

An integral domain R is termed quasicondensed if I^n = {i_1 i_2...i_n : i_j in I for 1 <= j <= n} for each positive integer n and each two-generated ideal I = (a,b) of R. R is said to be condensed if
IJ = {ij: i in I, j in J} for all ideals I and J of R. The main theorem shows that an integral domain is a Bezout domain if and only if it is integrally closed and condensed. An example (a D+M construction) is given of an integrally closed quasicondensed domain which is not a Bezout domain. Reviewed by Anne Grams

90e:13019 13F30 (13B20 13G05)
Gottlieb, Christian (S-STOC)
On condensed Noetherian domains whose integral closures are discrete valuation rings.
Canad. Math. Bull. 32 (1989), no. 2, 166-168.

Following D. F. Anderson and the reviewer [same journal 26 (1983), no. 1, 106-114; MR 84a:13019] an integral domain R is said to be condensed in case IJ = {ij : i in I, j in J} for all ideals I,J of R. The author defines an integral domain R to be strongly condensed if for every pair I,J of ideals of R, either IJ = aJ for some a in I or IJ = Ib for some b in J. Suppose henceforth that R is a Noetherian integral domain whose integral closure R' is a discrete valuation ring. It is proved that if R is condensed, then R contains an element of value 2 (in the associated discrete rank 1 valuation). It is not known whether the converse holds, nor whether all condensed domains are strongly condensed. As a partial converse, it is proved that R is strongly condensed under the following conditions: (R',M') is a finitely generated R-module, R'/M' is isomorphic to R/M and R contains an element of value 2. Reviewed by David E. Dobbs

1 955 608 13A15 (13Bxx)
Anderson, D. D.; Dumitrescu, Tiberiu
Condensed domains.
Canad. Math. Bull. 46 (2003), no. 1, 3-13.
http://journals.cms.math.ca/cgi-bin/vault/view/anderson8107

Abstract: An integral domain D with identity is condensed (resp., strongly condensed) if for each pair of ideals I,J of D, IJ = {ij : i in I, j in J} (resp., IJ = iJ for some i in I or IJ = Ij for some j in J). We show that for a Noetherian domain D, D is condensed if and only if Pic(D) = 0 and D is locally condensed, while a local domain is strongly condensed if and only if it has the two-generator property. An integrally closed domain D is strongly condensed if and only if D is a Bezout generalized Dedekind domain with at most one maximal ideal of height greater than one. We give a number of equivalencies for a local domain with finite integral closure to be strongly condensed. Finally, we show that for a field extension k < K, the domain D = k + XK[[X]] is condensed if and only if [K:k] <= 2 or [K:k] = 3 and each degree-two polynomial in k[X] splits over k, while D is strongly condensed if and only if [K:k] <= 2.