The prime numbers are numbers that are divisible by exactly two numbers. Fields are rings that have exactly two ideals. Moreover, the number of ideals of the product of two rings is the product of the number of ideals of both rings. A natural question one may hence ask is this:
Is every ring a product of fields?
In general, the answer is no, for the following reason: Obviously, the characteristic of a product of rings is the least common multiple of the characteristics of the factors. Hence, rings like $\mathbb Z / 4\mathbb Z$ are not achievable in this way. Moreover, the characteristic of any ring that is the products of fields will have the form $p_1 \cdots p_n$ (where $p_1, \ldots, p_n$ are distinct primes). Moreover, for noncommutative rings, we'll have to allow skew-fields. Hence, the question above becomes the following two questions:
- What is the class of rings that is the product of (skew-)fields?
- Is there a decomposition of arbitrary rings into (skew-)fields (that necessarily is not the product)?