A semisimple ring is a ring that can be decomposed into a direct sum of simple rings, meaning that it has no non-trivial two-sided ideals. This property is analogous to a semisimple group, which is a group without non-trivial normal subgroups.
A maximal ideal is an ideal that is not properly contained in any other ideal. A semisimple ring with a unique maximal ideal means that there is only one maximal ideal in the ring. This is significant because it implies that the ring is simple, and therefore has no non-trivial two-sided ideals.
A simple ring has no non-trivial two-sided ideals, while a semisimple ring with a unique maximal ideal has only one maximal ideal. This means that while both rings are simple, the latter has a unique maximal ideal that is not present in the former.
Yes, a semisimple ring can have more than one maximal ideal. However, a semisimple ring with a unique maximal ideal is a special case where there is only one maximal ideal in the ring.
Some examples of semisimple rings with a unique maximal ideal include the field of real numbers, the field of complex numbers, and the ring of quaternions. In general, any division ring (a ring where every non-zero element has a multiplicative inverse) is a semisimple ring with a unique maximal ideal.