Page 196 - DMTH403_ABSTRACT_ALGEBRA

Unit 19: The Field of Quotient Euclidean Domains

Another example of a prime ideal is Notes

Example: Let R be an integral domain. Show that I = ((0, x) | x  R) is a prime ideal of R x R.
Solution: Firstly, you know that I is an ideal of R x R. Next, it is a proper ideal since I  R x R.
Now, let us check if I is a prime ideal or not. For this let (a , b ), (a , b )  R x R such that (a , b )
2
2
1
1
1
1
(a , b )  I. Then (a a , b b ) = (0, x) for some x  R
2
1 2
1 2
2
 a a = 0, i.e., a, = 0 or a = 0, since R is a domain. Therefore, (a, b )  I or (a , b )  I. Thus, I is a
2
2
2
1 2
1
prime ideal.
Now we will, prove the relationship between integral domains and prime ideals.
Theorem 1: An ideal P of a ring R with identity is a prime ideal of R if and only if the quotient
ring R/P is an integral domain.
Proof: Let us first assume that P is a prime ideal of R. Since R has identity, so has R/P. Now, let
a + P and b + P be in R/P such that (ai – P) (b + P) = P, the zero element of R/P. Then ab+P = P, i.e.,
ab  P. As P is a prime ideal of R either a  P or b  P. So either a + P = P or b+P = P.
Thus, R/P has no zero divisors.
Hence, R/P is an integral domain.
Conversely, assume that R/P is an integral domain. Let a, b  R such that ab  P. Then ab + P =
P in R/P, i.e., (a + P) (b + P) = P in R/P. As R/P is an integral domain, either a + P = P or b + P =
P, i.e., either a E P or b  P. This shows that P is a prime ideal of R.
An ideal mZ of Z is prime iff m is a prime number. Can we generalise this relationship between
prime numbers and prime ideals in Z to any integral domain? To answer this let us first try and
suitably generalise the concepts of divisibility and prime elements.
Definition: In a ring R, we say that an elements divides an element b if b = ra for some r  R. In
this case we also say that a is a factor of b, or a is a divisor of b.

Thus, 3 divides 6 in Z , since 3.2  6.
7
Now let us see what a prime element is.
Definition: A non-zero element p of an integral domain R is called n prime element if
(i) p does not have a multiplicative inverse, and
(ii) whenever a, b  R and p | ab, then p | a or p | b.
Can you say what the prime elements of Z are? They are precisely the prime numbers and their
negatives.
Now that we know what a prime element is, let us see if we can relate prime ideals and prime
elements in an integral domain.
Theorem 2: Let R be an integral domain. A non-zero element p  R is n prime element if and
only if Rp is a prime ideal of R.
Proof: Let us first assume that p is a prime element in R. Since p does not have a multiplicative
inverse, 1  Rp. Thus, Rp is a proper ideal of R. Now let a, b  R such that ab  Rp. Then ab = rp
for some r  R
 p | a or p | b, since p is a prime element.
 a = xp or b = xp for some x  R.
a  p or b  Rp .
Thus ab  Rp  either a  Rp or b G  Rp, i.e., Rp is a prime ideal of R.

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