Group Theory: Multiplicative Cosets Explained

[Mr Davis 97]

In learning group theory, you learn about cosets as a partition of the entire group with respect to a subgroup. Since we are dealing with groups, there is only on operation that can be used to form the cosets. But when we come to rings, we now have two operations. So my question is, why do we only care about additive cosets, as opposed to defining multiplicative cosets and investigating them?

 

[member 587159]

In learning group theory, you learn about cosets as a partition of the entire group with respect to a subgroup. Since we are dealing with groups, there is only on operation that can be used to form the cosets. But when we come to rings, we now have two operations. So my question is, why do we only care about additive cosets, as opposed to defining multiplicative cosets and investigating them?

You can investigate multiplicative cosets as well. This is frequently done in group theory. I don't know why you should think that we only consider additive cosets.

 

[fresh_42]

In learning group theory, you learn about cosets as a partition of the entire group with respect to a subgroup. Since we are dealing with groups, there is only on operation that can be used to form the cosets. But when we come to rings, we now have two operations. So my question is, why do we only care about additive cosets, as opposed to defining multiplicative cosets and investigating them?

Because multiplication in a ring isn't a group. It is nothing more than an arbitrary binary operation. O.k. it's distributive and usually associative, but that's it.

However, if we look at the subsets $U \subseteq R$ of a ring, for which cosets usually are considered, we find, that $U$ is required to be an ideal. And as such $r\cdot U \subseteq U$ is required, i.e. the multiplication at least should behave nicely.

If we considered multiplicative cosets only, without being additive, then we would run into difficulties, as the ring multiplication isn't a group structure. First we will probably exclude $0$, because this element doesn't belong into multiplication and damages everything there. But what to do next with zero divisors? So let's say we have an integral domain. But what should a typical $U$ be? We don't have inverse elements, so is it any arbitrary set? Then what for should it be good to look at it? If we continue this way, by requiring new conditions if needed, we will probably end up with the multiplicative group of a division ring, which is just a multiplicative group that is already considered in the realm of group theory.

It is an enlightening exercise to try to define a well-defined structure on $\{r\cdot U\}$ and to see in the process, which properties are actually needed and what for. This holds also true for groups. It demonstrates the difference between an ordinary subgroup and a normal subgroup. In doing so, one can actually see, why this at first glance deliberate condition of normality actually comes into play!