Primes and Associates in Rings

In summary, the conversation discusses whether or not the statement "If a is prime and a, b are associates, then b is also prime" is true or false. The attempt at a proof involves showing that if b|xy, then b=au for some unit u. However, finding a counterexample proves to be difficult without knowing how to find units in a given ring. The conversation ends with the acknowledgement of successfully solving the problem.
  • #1
Zoe-b
98
0

Homework Statement


Let a, b be members of a commutative ring with identity R. If a is a a prime and a, b are associates then b is also prime. True/False


Homework Equations


Definitions: a is prime if a|xy implies a|x or a|y
a and b are associates if there exists a unit u s.t a=bu


The Attempt at a Solution


First off I'm not sure whether its even true or not; so I've tried to both prove it and find a counter example:
Attempt at a proof:
let b|xy.
Then if b = au for some unit u, au|xy, that is for some k in R auk = xy.
so ak divides xy(u^(-1))
but k does not necessarily have an inverse so this doesn't really get me anywhere.

I've shown in an earlier question that if a is irreducible and a, b are associates then b is irreducible. Prime implies irreducible but not vice versa so if there is a counter example it will depend on an element of R that is irreducible but not prime. I know examples of these can be found in rings such as Z(5i) (the set of numbers a+5bi for a, b in Z). But I don't know how to find units of this ring.
Consider (a+5bi)(c+5di) = 1.
This gives the simultaneous equations 1+25bd-ac = 0 = ad+bc
But so far I have not managed to find any integer solutions (and to be honest I don't know any way of doing this other than trial and error).

Thanks for any help!
 
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  • #2
Zoe-b said:
First off I'm not sure whether its even true or not; so I've tried to both prove it and find a counter example:
Attempt at a proof:
let b|xy.
Then if b = au for some unit u, au|xy, that is for some k in R auk = xy.

OK, so stop here. We know that auk=xy. So this implies that [itex]ak=u^{-1}xy[/itex]. Thus a divides [itex]u^{-1}xy[/itex]. Now use that a is prime.
 
  • #3
Thank you! Done it now :)
 

What are "Primes and Associates in Rings"?

"Primes and Associates in Rings" is a mathematical concept that deals with prime numbers and their relationships within a ring structure. This concept is commonly studied in abstract algebra and number theory.

What is a prime number?

A prime number is a positive integer that is only divisible by 1 and itself. In other words, it has exactly two factors.

What is a ring?

A ring is a mathematical structure that consists of a set of elements and two operations (usually addition and multiplication) that satisfy specific properties. Examples of rings include the set of integers and the set of polynomials.

What is the role of primes in rings?

Primes play a significant role in rings as they are used to define irreducible elements and prime ideals. They also help in factorization and understanding the structure of the ring.

How can "Primes and Associates in Rings" be applied in real-world scenarios?

The concept of "Primes and Associates in Rings" has many applications in various fields, including cryptography, coding theory, and computer science. For example, prime numbers are used in encryption algorithms to ensure secure communication over the internet.

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