## Semiring of Ideals

**Point of Post: **In this post we define the notion of a semiring and then define a semiring structure on the set of ideals of a ring .

*Motivation*

In our last post we saw that every ring has associated to it a complete modular lattice which is comprised of its ideals. More particularly, we saw that there was a way to ‘add’ ideals together which was based off the addition of the ring itself. An obvious question then is whether or not we can figure out a way to ‘multiply’ ideals together which uses the multiplication of a ring. But then! We think “we’ve got a set with a multiplication and addition…that sounds like…well..you know“. So we pursue this and see if these new notions of addition and multiplication of ideals make into a ring. Sadly, this isn’t the case. But, we get pretty darn close. We find out that with these operations is a s*emiring *which is roughly a ring where the additive structure is no longer an abelian group but a commutative monoid.

*Product of Ideals*

Let be a ring. We define the *multiplication *of two ideals , denoted by concatenation, to be the ideal . In other words, the product of two ideals is the ideal generated by the setwise product of the ideals. But, just as the case with the sum of ideals this formidable looking definition reduces very nicely. Namely:

**Theorem: ***Let be a ring and . Then, .*

**Proof: **Denote the right hand side of the above equation as . Evidently since it’s obviously a subgroup and left or right multiplying by a general -element on a finite sum of the form is just absorbed by the ‘s on the left and the ‘s on the right. Since it clearly contains the set we have that . Conversely, is evidently a subset of since it contains all elements of the form $late ab$ with and and thus must contain all finite sums of that type.

In fact, from this it’s easy to see that if we define to be equal to then

.

So an obvious question is how this product of ideals relates to former operations on ideals such as intersection and sum. Unsurprisingly, it distributes over addition but perhaps slightly unexpected (not really if you think about it but viscerally kind of weird) the product of ideals is contained in the intersection of the ideals. Indeed:

**Theorem: ***Let be a ring and . Then:*

*where we’ve used to denote the zero ideal . Furthermore, if is commutative then*

**Proof: ** is obvious.

To see we let then evidently for some , and . But, then and so . Conversely, we note that evidently (since we can consider sums of the form and ) and thus since it’s a subgroup we have that . The other result is similar.

To see we merely note that if and then (by right absorption) and (by left absorption) so that . Thus, is an ideal of containing the set of all -elements of the form so that, by definition, .

To see we merely note that if then where , , and . But then, but evidently so . Since was arbitrary the conclusion follows.

*Semiring of Ideals*

We now show the true extent of the interaction between the product and sum of ideals. To put it in the right light, we give the ‘structure’ with the sum and product of ideals takes on a name. Namely, a set is called a *semiring *if is a commutative monoid with identity , is a semiring, left and right distributes over , and finally . What we now claim is that with the sum and product of ideals forms a semiring. Indeed:

**Theorem: ***Let be a ring, then forms a semiring with additive identity .*

**Proof: **To see that is a commutative monoid is easy since evidently the addition of ideals inherits associativity from the associativity of the -addition, it similarly inherits the commutativity from the -addition, and evidently .

It’s easy to see that is a semiring since the product of ideals inherits associativity from the -product.

The last two properties, left and right distributivity as well as left and right annhilation by were proven in the last theorem.

**References:**

1. Dummit, David Steven., and Richard M. Foote. *Abstract Algebra*. Hoboken, NJ: Wiley, 2004. Print.

2. Bhattacharya, P. B., S. K. Jain, and S. R. Nagpaul. *Basic Abstract Algebra*. Cambridge [Cambridgeshire: Cambridge UP, 1986. Print.

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