List all zero-divisors in Z_20. Can you see relationship between the zero-divisors of Z_20 and the units of Z_20?

snowlovelydayM 2020-11-30 Answered
List all zero-divisors in Z20. Can you see relationship between the zero-divisors of Z20 and the units of Z20?
You can still ask an expert for help

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Solve your problem for the price of one coffee

  • Available 24/7
  • Math expert for every subject
  • Pay only if we can solve it
Ask Question

Expert Answer

falhiblesw
Answered 2020-12-01 Author has 97 answers

Assume that R be a commutative ring and a be a nonzero element of R.
Zero-divisors An element a of a ring R is called a zero divisor if there exists a nonzero x such that ax = 0.
From the definition of zero divisors, find the zero divisors of Z20 in the following.
Since, Z20={0,1,2,.,19}
2×10=0, Since 20,100
4×4=0, Since 40,50
4×15=0, Since 40,150
8×5=0, Since 80,50
12×5=0, Since 120,50
6×10=0, Since 60,100
8×10=0, Since 80,100
14×10=0, Since 140,100
16×10=0, Since 160,100
18×10=0, Since 180,100
Therefore, zero divisors of Z20 are 2, 4, 5, 6, 8, 10, 12, 14, 15, 16 and 18.
A unit in a ring is an element u such that there exists u1 where u.u1=1
Now find the units of Z20 in the following.
Since the elements which are relatively prime to 20 is called units.
Therefore, the relatively primes to 20 are 1, 3, 7, 9, 11, 13, 17, and 19.
Then,
Units of 1=1, Since 1×1=1
Units of 3=7, Since 3×7=1
Units of 7=3, Since 7×3=1
Units of 9=9, Since 9×9=1
Units of 11=11, Since 11×11=1
Units of 13=17, Since 13×17=1
Units of 19=19, Since 19×19=1
Hence, units are 1, 3, 7, 9, 11, 13, 17, 19.
These units cannot be zero-divisors.

Not exactly what you’re looking for?
Ask My Question

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Relevant Questions

asked 2021-06-14
i don't get the ideas to solve this question. can anyone help me?
asked 2020-11-27
Let R be a commutative ring. If I and P are idelas of R with P prime such that I!P, prove that the ideal P:I=P
asked 2022-01-07
Let * be a binary operation on the set of real numbers R defined as follows:
ab=a+b3(ab)2, where a,bR
- Prove that * is commutative but not associative algebraic operation on R.
- Find the identity element for * .
- Show that 1 has two inverses with respect to *.
asked 2022-06-20
Spectral radius in Banach algebra is commutative
I want to show that for a Banach algebra A and elements x , y A, we have
r A ( x y ) = r A ( y x ) ,
where r A is a spectral radius. This is how I am trying to do that:
r a ( x y ) = lim n ( x y ) n 1 n = lim n x ( y x ) n 1 y 1 n .
And this is where I stuck, since I only know that x y x y . Could you please suggest any ideas on how to proceed with the proof?
asked 2022-06-08
Is every finite ring a matrix algebra over a commutative ring?
- Can every finite ring R be written as a subring of Mat n × n A for some commutative ring A?
- If not, then what is/are the smallest ring(s) that cannot be?
asked 2022-06-01
Let X be a compact hausdorff topological space with more than one element.Then prove that the ring C ( X ) of complex valued continuous functions on X is not an integral domain. Thanks for any help. Actually this question arose when I was trying to prove that any commutative C -algebra which is also an integral domain must be isomorphic to C and I think this statement is correct. The problem I am having is with the case when X is connected.
asked 2022-05-24
Let R be a commutative finite dimensional K-algebra over a field K (for example the monoid ring of a a finite monoid over a field). Assume we have R in GAP. Then we can check whether R is semisimple using the command RadicalOfAlgebra(R). When the value is 0, R is semisimple. Thus R can be written as a finite product of finite field extensions of K.
Question: Can we obtain those finite field extensions of K or at least their number and K-dimensions using GAP?

Solve your problem for the price of one coffee

  • Available 24/7
  • Math expert for every subject
  • Pay only if we can solve it
Ask Question