already knowing by definition a prime number is any number greater than 1 with exactly 2 factors, 1 and itself, and taking into account that 1 is a unit, 2 is a prime, and 4 is a composite all by definition
2007-01-08 14:23:22 · 5 answers · asked by j b 2 in Science & Mathematics ➔ Mathematics
The integers are not the only mathematical system that we work with, there are many generalizations referred to as Rings. In a ring, the definition of "prime" only makes sense for non-units. Primes behave one way and units behave another way. Being a unit overrides primeness.
More specifically:
In a commutative unital ring R, if p is a prime then R/
is an integral domain. However if u is a unit, then R/ is the zero ring. ( Here indicates the ideal generated by the element a. )
2007-01-08 23:21:51 · answer #1 · answered by AnyMouse 3 · 0⤊ 0⤋
This is pretty much what you said, but a prime number must be divisible by 2 numbers, 1 and itself, and 1 is only divisible by 1 factor (that number being both 1 and itself)
2007-01-08 14:27:58 · answer #2 · answered by twistedangel 3 · 0⤊ 0⤋
My understanding is that some of the deeper theorems on prime numbers would not work if 1 were considered prime, and this is a key reason the primes are defined the way they are.
2007-01-08 15:20:16 · answer #3 · answered by Edward W 4 · 0⤊ 0⤋
Prime numbers must have exactly two different factors and 1 only has one factor.
2007-01-08 14:27:19 · answer #4 · answered by hayharbr 7 · 0⤊ 0⤋
1 is a unit, it is the only number such that , if you multiply with any other number a, nothing changes.
2007-01-08 17:30:52 · answer #5 · answered by Theta40 7 · 0⤊ 1⤋