55 798
Assignments Done
97,2%
Successfully Done
In December 2017
Your physics homework can be a real challenge, and the due date can be really close — feel free to use our assistance and get the desired result.
Be sure that math assignments completed by our experts will be error-free and done according to your instructions specified in the submitted order form.
Our experts will gladly share their knowledge and help you with programming homework. Keep up with the world’s newest programming trends.

Answer on Abstract Algebra Question for Annemarie

Question #4996
Prove that 5Z is both a prime and maximal ideal of Z
Expert's answer
Let& R be a ring. A two sided ideal I of R is called Maximal Ideal if I is not equal to R and no proper ideal of R properly contains I.
For suppose that I is an ideal of Z properly containing& 5I, then there exists some m ε I but m does not in 5I, i.e. 5 does not divide m. Then gcd(5,m)=1 , since& 5 is prime.
Therefore, we can write
1= 5s + m t
for integers s and t. Since 5s& in I& and mt is also in I, this means 1 in I& and hence I = Z.
Therefore, 5Z is a maximal ideal.
Let R be a commutative ring. An ideal I of R is called prime if I not eqal to R. And whenever ab in I for some a, b in R, then either a in I or b in I.
The ideal 5Z is prime in Z, since the product of two integers is a multiple of 5 only if at least one of the two is a multiple of 5.

Need a fast expert's response?

Submit order

and get a quick answer at the best price

for any assignment or question with DETAILED EXPLANATIONS!

Comments

No comments. Be first!

Leave a comment

Ask Your question