It depends on how your textbook or lecturer defines "weakly prime ideal" as this isn't standard terminology. What I _suspect_ is meant here is that we typically have two results: Result 1. If R is a commutative ring and J is an ideal of R, then J is a prime ideal if and only if R/J is an integral domain. Result 2. If R is a commutative ring and x is an element of R, then x is a prime element if and only if is a prime ideal. The way I stated the two results is actually erroneous, since we do not consider 0 to be a prime element of a ring. So if we hold result 2 is written correctly, there can never be a situation in which is a prime ideal. However, if R is an integral domain, then R/ is isomorphic to R, so R/ is an integral domain, meaning that is a prime ideal by result 1. There are two ways to patch up this contradiction. Either you put x not equal to 0 in result 2, or you put J a nonzero ideal in result 1. In my experience, typically people modify result 2 and write it as "If R is a commutative ring and x is a _nonzero_ element of R, then x is a prime element if and only if is a prime ideal." However, one could modify result 1 and say "If R is a commutative ring and J is a _nonzero_ ideal of R, then J is a prime ideal if and only if R/J is an integral domain." But seeing how the ideal behaves "like" prime ideals inside of integral domains, maybe your textbook or lecturer then calls it "weakly prime" since they don't want to consider it prime. But as I said before, most often, I have seen people say that is simply a prime ideal in integral domains.
It seems I jumped the gun a bit. I just looked up "weakly prime ideal". It was an idea introduced around 2003, it seems. Let R be a commutative ring. A proper ideal P is called a "weakly prime ideal" of R if whenever a _nonzero_ product ab is in P, then a is in P or b is in P. This is very close to the definition of a prime ideal. The only difference is that prime ideals do not require the product ab to be _nonzero._ So weakly prime ideals are very close to prime ideals, but if a product is 0, we don't require one of the factors to be in the ideal. While the ideal is prime in any integral domain, the ideal is _weakly_ prime in any commutative ring, vacuously. Because there are no nonzero products in , so the condition that whenever a nonzero product ab is in , then a is in or b is in is _vacuously_ satisfied.
Good stuff, thanks.
sir i really help alot in algebra ..thank u sir ple take more example
Sir can you please explain why is a weakly prime ideal
It depends on how your textbook or lecturer defines "weakly prime ideal" as this isn't standard terminology.
What I _suspect_ is meant here is that we typically have two results:
Result 1. If R is a commutative ring and J is an ideal of R, then J is a prime ideal if and only if R/J is an integral domain.
Result 2. If R is a commutative ring and x is an element of R, then x is a prime element if and only if is a prime ideal.
The way I stated the two results is actually erroneous, since we do not consider 0 to be a prime element of a ring. So if we hold result 2 is written correctly, there can never be a situation in which is a prime ideal. However, if R is an integral domain, then R/ is isomorphic to R, so R/ is an integral domain, meaning that is a prime ideal by result 1.
There are two ways to patch up this contradiction. Either you put x not equal to 0 in result 2, or you put J a nonzero ideal in result 1. In my experience, typically people modify result 2 and write it as "If R is a commutative ring and x is a _nonzero_ element of R, then x is a prime element if and only if is a prime ideal." However, one could modify result 1 and say "If R is a commutative ring and J is a _nonzero_ ideal of R, then J is a prime ideal if and only if R/J is an integral domain."
But seeing how the ideal behaves "like" prime ideals inside of integral domains, maybe your textbook or lecturer then calls it "weakly prime" since they don't want to consider it prime.
But as I said before, most often, I have seen people say that is simply a prime ideal in integral domains.
It seems I jumped the gun a bit. I just looked up "weakly prime ideal". It was an idea introduced around 2003, it seems.
Let R be a commutative ring. A proper ideal P is called a "weakly prime ideal" of R if whenever a _nonzero_ product ab is in P, then a is in P or b is in P.
This is very close to the definition of a prime ideal. The only difference is that prime ideals do not require the product ab to be _nonzero._ So weakly prime ideals are very close to prime ideals, but if a product is 0, we don't require one of the factors to be in the ideal.
While the ideal is prime in any integral domain, the ideal is _weakly_ prime in any commutative ring, vacuously. Because there are no nonzero products in , so the condition that whenever a nonzero product ab is in , then a is in or b is in is _vacuously_ satisfied.
nice :) thank you
Good.
so 0 is a prime number? xd
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