Prime Ideals For Commutative Rings
An ideal P of a commutative ring R is prime if it has the following two properties:
- If a and b are two elements of R such that their product ab is an element of P, then a is in P or b is in P,
- P is not equal to the whole ring R.
This generalizes the following property of prime numbers: if p is a prime number and if p divides a product ab of two integers, then p divides a or p divides b. We can therefore say
- A positive integer n is a prime number if and only if the ideal nZ is a prime ideal in Z.
Read more about this topic: Prime Ideal
Famous quotes containing the words prime, ideals and/or rings:
“The Prime Minister has an absolute genius for putting flamboyant labels on empty luggage.”
—Aneurin Bevan (1897–1960)
“The nineteenth century was completely lacking in logic, it had cosmic terms and hopes, and aspirations, and discoveries, and ideals but it had no logic.”
—Gertrude Stein (1874–1946)
“You held my hand
and were instant to explain
the three rings of danger.”
—Anne Sexton (1928–1974)