In abstract algebra, a non-zero non-unit element in an integral domain is said to be irreducible if it is not a product of two non-units.
Irreducible elements should not be confused with prime elements. (A non-zero non-unit element a in a commutative ring R is called prime if whenever a | bc for some b and c in R, then a|b or a|c.) In an integral domain, every prime element is irreducible, but the converse is not true in general. The converse is true for UFDs (or, more generally, GCD domains.)
Moreover, while an ideal generated by a prime element is a prime ideal, it is not true in gener...
In proving that all non-zero non-units of a PID are a product of irreducibles, theres:
"We now show that $a$ is a product of irreducibles. If $a$ is irreducible,
we are done. Otherwise let $p_1$ be an irreducible such that $p_1|a$. Then
$a=p_1c_1$. If $c_1$ is a unit, we are done. "
How...In the topology of metric spaces the Heine–Borel theorem, named after Eduard Heine and Émile Borel, states:
For a subset S of Euclidean space Rn, the following two statements are equivalent:
*S is closed and bounded
*every open cover of S has a finite subcover, that is, S is compact.
In the context of real analysis, the former property is sometimes used as the defining property of compactness. However, the two definitions cease to be equivalent when we consider subsets of more general metric spaces and in this generality only the latter property is used to define compactness. In fact, the...
