These are my answers to a past exam question, just wondering if someone could please check whether they are right and also whether I have written enough to gain the full marks for each question?

Thanks in advance!

**Is 3 + 15Z a zero divisor on R? [3 marks]**
Yes because 3.5 = 0.

__More precisely:__ (3+15Z)(5+15Z) = 15+15Z = 0 + 15Z **What is the inverse of 7 + 15Z in R? [3 marks]**
The inverse is 13 + 15Z because (7 + 15Z)(13 + 15Z) = 1.

**Well Done**

**Is 7 + 15Z a zero divisor in R? [3 marks]**
No because there are no such elements that a.7 = 0, where a is in R.

Generall__y:__ "If an inverse exists for an element a, then it cannot be a zero divisor". If a.b = 0(with a and b not zero) and a^{-1} exists, then a^{-1}(a.b) = a^{-1}.0 => b = 0. Contradiction. **Is R a field? [2 marks]**
No because not every non-zero element has an inverse in R.

**Well Done. You should also illustrate a counter example. So you can write "No because not every non-zero element has an inverse in R. ****For instance, 3 does not have an inverse".**

**Find an ideal I of R consisting of 3 elements. [4 marks]**
An ideal of R consisting of 3 elements is {0, 5, 10}.

**What is the number of elements of R/I? [2 marks]**
R/I = 15/{0,5,10} = 15/3 = 5 elements.

__More precisely:__ [0] = {0,5,10},[1] = {1,6,11}, [2] = {2,7,12}, [3] = {3,8,13},[4] = {4,9,14}.

**Write down the multiplication table of R/I. [4 marks]**
X 0 1 2 3 4

0 1 1 2 3 4

1 0 2 4 1 3

2 0 2 4 1 3

4 0 4 3 2 1

**Its wrong.The table should look like this:**

X [0] [1] [2] [3] [4]

[0] 0 0 0 0 0

[1] 0 1 2 3 4

[2] 0 2 4 6 8

[3] 0 3 6 9 12

[4] 0 4 8 12 1 **But 6 belongs to [1], 8 belong to [3], 12 belongs to [2] and 9 belongs to [4]**
Thus:

** X [0] [1] [2] [3] [4]**

[0] 0 0 0 0 0

[1] 0 1 2 3 4

[2] 0 2 4 1 3

[3] 0 3 1 4 2

[4] 0 4 3 2 1

A Note: The point of this exercise was to convince you that R/I is a multiplicative group. Can you see it?