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Handout Friday 10/31
We have been spending a lot of time considering cosets of groups and forming quotient groups. What about rings?
As an example, we have the ring \(\Z\text{,}\) which is also an abelian group, and we formed the cosets \(\Z/5\Z\text{,}\) which we said was itself a group. But \(\Z\) is also a ring: multiplication makes sense. Can we also multiply cosets?
Back up a bit. What do cosets for rings even mean? We want to have a subring \(S\) (which is also a subgroup) of the ring \(R\text{.}\) Do we form a coset with \(a+S\) or \(aS\text{.}\) See why the latter doesn’t make sense.
Great, so we have cosets. Can we add and multiply them? Note that since the group part of a ring is always abelian, the subgroup will automatically be normal. So there will be no problem with coset addition. What about multiplication?
If we consider \((a+S)(b+S)\text{,}\) we want this to be \(ab+S\text{.}\) What are things inside \((a+S)(b+S)\text{?}\) It is some \((a+s_1)(b+s_2)\text{.}\) Simplify this.
What we really need is for \(as_2+bs_1 + s_1s_2\) to be in \(S\text{.}\) Is it? Well it will be if \(S\) is an ideal.
An ideal is a subring that is closed under multiplication by elements outside the ideal.
An easy way to get an ideal in a ring is to look at \(\langle a \rangle\text{,}\) the set of all multiples of \(a\text{.}\) We call this the ideal generated by \(a\text{.}\)
