The ideal product is well-defined

In TAN, the product of two finitely generated ideals (A) and (B) was defined to be (ab \ |\ a \in A, b \in B). Argue that this is a well-defined operator on the set of ideals of a fixed ring R.


More broadly, IJ = (xy \ |\ x \in I, y \in J). If I = (A) and J = (B), then each x is an R-linear combination of the a_i \in A, and each y a combination of the b_i \in B. Since R is a commutative ring, then, each xy is an R-linear combination of a_ib_j where a_i \in A and b_j \in B; that is, IJ \subseteq (ab \ |\ a \in A, b \in B). The reverse inclusion follows from this previous exercise.

A more modern treatment would derive the generating set characterization of IJ from the more general definition.

Post a comment or leave a trackback: Trackback URL.

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s

%d bloggers like this: