The Unapologetic Mathematician

Mathematics for the interested outsider

Semigroup rings

Today I’ll give another great way to get rings: from semigroups.

Start with a semigroup S. If it helps, think of a finite semigroup or a finitely-generated one, but this construction doesn’t much care. Now take one copy of the integers \mathbb{Z}_s for each element s of S and direct sum them all together. There are two ways to think of an element of the resulting abelian group, as a function f:S\rightarrow\mathbb{Z} that sends all but finitely many elements of S to zero, or as a “formal finite sum” c_1e_{s_1}+c_2e_{s_2}+...+c_ne_{s_n} where each c_i is an integer and e_s is “1” from the copy of \mathbb{Z} corresponding to s.

I’ll try to talk in terms of both pictures since some people find the one easier to understand and some the other. We can go back and forth by taking a valid function and using its nonzero values as the coefficients of a formal sum: f=\sum\limits_{s\in S}f(s)e_s. This sum is finite because most of the values of f are zero. On the other hand, we can use the coefficients of a formal sum to define a valid function.

So we’ve got an abelian group here, but we want a ring. We use the semigroup multiplication to define the ring multiplication. In the formal sum picture, we define e_{s_1}e_{s_2}=e_{s_1s_2}, and extend to sums the only way we can to make the multiplication satisfy the distributive law. In the function picture we define \left[fg\right](s)=\sum\limits_{xy=s}f(x)g(y) where we take the sum over all pairs (x,y) of elements of S whose product is s. This takes the product of all nonzero components of f and g and collects the resulting terms whose indices multiply to the same element of the semigroup.

The ring we get is called the “semigroup ring” of S, written \mathbb{Z}[S]. There are a number of easy variations on the same theme. If S is actually a monoid we sometimes say “monoid ring”, and note that the ring has a unit given by the identity of the monoid. If S is a group we usually say “group ring”. If in any of these cases we start with a commutative semigroup (monoid, group) we get a commutative ring.

So here’s the really important thing about semigroup rings. If we take any ring R and forget its additive structure we’re left with a semigroup. If we take any semigroup homomorphism from S to this “underlying semigroup” of R we can uniquely extend it to a ring homomorphism from \mathbb{Z}[S] to R. This is just like what we saw for free groups, and it’s just as important.

As a side note, I want to mention something about the multiplication in group rings. Since xy=s only if y=x^{-1}s we can rewrite the product formula in the function case \left[fg\right](s)=\sum\limits_{x\in S}f(x)g(x^{-1}s). This way of multiplying two functions on a group is called “convolution”, and it shows up all over the place.

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April 12, 2007 - Posted by | Ring theory

9 Comments »

  1. [...] Now we come to a really nice example of a semigroup ring. Start with the free commutative monoid on generators. This is just the product of copies of the [...]

    Pingback by Polynomials « The Unapologetic Mathematician | April 16, 2007 | Reply

  2. [...] Free Ring on an Abelian Group Last week I talked about how to make a ring out of a semigroup by adding an additive structure. Now I want to do the other side. Starting with an abelian group [...]

    Pingback by The Free Ring on an Abelian Group « The Unapologetic Mathematician | April 19, 2007 | Reply

  3. [...] can also start with any semigroup and build the semigroup algebra just like we did for the semigroup ring . As a special case, we can take to be the free commutative monoid on generators and get the [...]

    Pingback by Algebras « The Unapologetic Mathematician | May 8, 2007 | Reply

  4. [...] representation properly extends that of a group representation. Given any group we can build the group algebra . As a vector space, this has a basis vector for each group element . We then define a [...]

    Pingback by Algebra Representations « The Unapologetic Mathematician | October 24, 2008 | Reply

  5. [...] Group Algebra A useful construction for our purposes is the group algebra . We’ve said a lot about this before, and showed a number of things about it, but most of [...]

    Pingback by The Group Algebra « The Unapologetic Mathematician | September 14, 2010 | Reply

  6. [...] it’s just a formal linear combination of singular -cubes. That is, for each we build the free abelian group generated by the singular -cubes in [...]

    Pingback by Chains « The Unapologetic Mathematician | August 5, 2011 | Reply

  7. Is there a standard term for a “reduced semigroup ring” when the semigroup has a zero? For a semigroup with zero element, the “full semigroup ring” can be taken quotient of by the ideal generated by the semigroup zero.

    Comment by Alexey Muranov | July 9, 2014 | Reply

  8. I don’t know that there is such a standard term.

    Comment by John Armstrong | July 9, 2014 | Reply

    • I was curious about “reduced semigroup rings” because matrix algebras are such. Consider a ring $R$ and a semigroup $S$ of $n^2+1$ elements $0$ and $(i,j)$, where $i$ and $j$ are taken from some $n$-element set, and the semigroup product is defined by $(i,j)(j,k)=(i,k)$, and $(i,j)(k,l)=0$ if $j\ne k$. Then the “reduced semigroup ring” of $S$ over $R$ is isomorphic to the ring of $n\times n$ matrices over $R$.

      Comment by Alexey Muranov | July 10, 2014 | Reply


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