1. ## A Ring's zero-divisors

Let $\displaystyle R = \mathbb{Z}_4 \oplus 3 \mathbb{Z}_{15}$ be the direct sum of the rings $\displaystyle \mathbb{Z}_4$ and $\displaystyle 3 \mathbb{Z}_{15}$, where $\displaystyle 3 \mathbb{Z}_{15} =\{ 0, 3, 6, 9, 12 \}$ is the subring of $\displaystyle \mathbb{Z}_{15}$.

What is the zero element of R? What is the unity of R? Also write down all of the zero-divisors of R.

My Attempt:

I think by 'zero element of R', they mean the identity of the abelian group (R, +) which is (0,0).

I think the unity is the identity under multipication and I think it is (1,1).

Am I right?

For the last question: we know that nonzero elements r, s of R are called zero-divisors if rs = 0. So the elements of R are:

(0,0), (0,3), (0,6), (0,9), (0,12)
(1,0), (1,3), (1,6), (1,9), (1,12)
(2,0), (2,3), (2,6), (2,9), (2,12)
(3,0), (3,3), (3,6), (3,9), (3,12)

Let r=(a,b) and s=(c,d). Then rs= (a,b)(c,d)=(0,0) iff b=d=0 so we have 3 possibilities: {(1,0),(2,0),(3,0)}, and the only one which works is (2,0) (because 2.2 mod 4 =0). Therefore (2,0) is the only zero-divisor. Is this correct?

2. Originally Posted by demode
Let $\displaystyle R = \mathbb{Z}_4 \oplus 3 \mathbb{Z}_{15}$ be the direct sum of the rings $\displaystyle \mathbb{Z}_4$ and $\displaystyle 3 \mathbb{Z}_{15}$, where $\displaystyle 3 \mathbb{Z}_{15} =\{ 0, 3, 6, 9, 12 \}$ is the subring of $\displaystyle \mathbb{Z}_{15}$.

What is the zero element of R? What is the unity of R? Also write down all of the zero-divisors of R.

My Attempt:

I think by 'zero element of R', they mean the identity of the abelian group (R, +) which is (0,0).

I think the unity is the identity under multipication and I think it is (1,1).

Am I right?

For the last question: we know that nonzero elements r, s of R are called zero-divisors if rs = 0. So the elements of R are:

(0,0), (0,3), (0,6), (0,9), (0,12)
(1,0), (1,3), (1,6), (1,9), (1,12)
(2,0), (2,3), (2,6), (2,9), (2,12)
(3,0), (3,3), (3,6), (3,9), (3,12)

Let r=(a,b) and s=(c,d). Then rs= (a,b)(c,d)=(0,0) iff b=d=0 so we have 3 possibilities: {(1,0),(2,0),(3,0)}, and the only one which works is (2,0) (because 2.2 mod 4 =0). Therefore (2,0) is the only zero-divisor. Is this correct?

Looks fine to me, but note that "rs= (a,b)(c,d)=(0,0) iff b=d=0" isn't correct; what is correct is "if rs = 0 then b=d=0".

Tonio

3. Even the first part is correct?

Also, I have another question. The characteristic char(R) of R is therefore (0,0), right? Because the least positive integer n having the property that nr = 0 for all elements r of R is the characteristic of R. Here I don't see any such positive integer, so the characteristic of R is (0,0)?

4. Originally Posted by demode
Even the first part is correct?

Also, I have another question. The characteristic char(R) of R is therefore (0,0), right? Because the least positive integer n having the property that nr = 0 for all elements r of R is the characteristic of R. Here I don't see any such positive integer, so the characteristic of R is (0,0)?

Note that the characteristic is NOT the least positive integer that...etc., lest zero wouldn't be characteristic ever since zero is not a positive integer

Tonio

5. Originally Posted by tonio

Note that the characteristic is NOT the least positive integer that...etc., lest zero wouldn't be characteristic ever since zero is not a positive integer

Tonio
Well, this is the exact definition of characteristic from my textbook:

"The least positive integer n with the property that nr = 0 for (all r of a ring R) is
called the characteristic of R. If no such positive integer exists, then the characteristic of R is 0."

So, what would be the characteristic in this problem?

6. Originally Posted by demode
Well, this is the exact definition of characteristic from my textbook:

"The least positive integer n with the property that nr = 0 for (all r of a ring R) is
called the characteristic of R. If no such positive integer exists, then the characteristic of R is 0."

So, what would be the characteristic in this problem?

But zero is not a POSITIVE integer! Anyway, since $\displaystyle 60r=0\,\,\forall r\in R$ then the characteristic is a divisor of 60...

Tonio

7. Originally Posted by tonio
But zero is not a POSITIVE integer! Anyway, since $\displaystyle 60r=0\,\,\forall r\in R$ then the characteristic is a divisor of 60...

Tonio
Tonio, the divisors of 60={1,2,3,4,5,6,10,12,15,20,30,60}. These are all positive integers, how do you know which one would work for ALL r in R?

8. Originally Posted by demode
Tonio, the divisors of 60={1,2,3,4,5,6,10,12,15,20,30,60}. These are all positive integers, how do you know which one would work for ALL r in R?

By inspection is one way, but it is weary. Pay attention that the characteristic MUST work both for 3 and 5....this already cuts sharply the possible options.

Tonio