Question

Give an example of a ring in which the cancellation law does not hold,that is find a ring R with nonzero elements a,x,y such that ax=ay but x #y (please be sure to provide a specific example in the ring you found

Answer 1

please help

Answer 2

Well, what are a few examples of rings?

Answer 3

what do you mean?

Answer 4

I am not sure of this question

Answer 5

Do you know the definition of a ring?

Answer 6

im confused

Answer 7

I don't quite understand -- I assume you are in some abstract algebra course?

Answer 8

yes right

Answer 9

Do you know the definition of a ring?

Answer 10

yes

Answer 11

Okay, so give me a few examples of rings.

Answer 12

ring how has at least two elements and has all the properties without mult invers ,mult identity and mult commit vie

Answer 13

I have no idea what that sentence is meant to say, I'm afraid. Just give me one example of a ring.

Answer 14

and a commitive ring is ring whose multiplication is commutative

Answer 15

Not definitions, examples.

Answer 16

okay

Answer 17

I just have that R(X) AND F(X) IS RING

Answer 18

I DONT HAVE EXAMPLE JUST DEFINATION

Answer 19

Well they don't do much good if you don't understand what the words mean. Do you understand what a group is?

Answer 20

I really understand ring what is mean but I do not understand whats the question mean

Answer 21

The question is asking for an example of a ring with elements a,b, and c such that
\[ a\otimes b = a\otimes c \]
but \[ b \neq c \]

Answer 22

And respectfully, if you can't give an example of a ring, then you don't know what a ring is.

Answer 23

yes because I do not have example of that and I asked because I do not know about it just know the definition

Answer 24

Well, like I asked before, do you know what a group is?

Answer 25

And can you give me an example of a group?

Answer 26

i think z7 is aring

Answer 27

z mode 7

Answer 28

wtf is this?

Answer 29

Yes, it is. Z / 7 is a ring. So here's the question: Does every multiplication have an inverse?

Answer 30

no

Answer 31

Okay, which ones are not invertible?

Answer 32

you ask about z mode 7 or what

Answer 33

Yes.

Answer 34

because its integer and there is no invers

Answer 35

Okay. Let's say I have an integers "a", "b", and "c".

Is it possible that a*b = a*c, with b not equal to c?

Answer 36

you mean a times b ?

Answer 37

okay i understood

Answer 38

Yes

Answer 39

yes that is possible of course

Answer 40

like 4*4=8*2

Answer 41

but what is the question want find example in ring then we have to prove does not equal

Answer 42

Notice that I'm saying
\[ a\otimes b = a\otimes c \]
the integer "a" is common to both terms.

Answer 43

ohhh

Answer 44

does have to be integer

Answer 45

yes, z/7, remember?

Answer 46

yes

Answer 47

i am really confused about that

Answer 48

About what?

Answer 49

about how can i solve it!

Answer 50

There are many very simple examples. Consider the ring of integers.

\[0 \otimes 4 = 0 \otimes 7 \]
right?

Answer 51

but the question say with nonzero elements

Answer 52

I understand, we're getting there.

Answer 53

So we can't find elements like that in the set of integers. What's another simple example? How about matrices?

Answer 54

iHave idea

Answer 55

if we take (1)mod 7*(6)mode 7=(1)mode 7 *(13) when we 13-7=6

Answer 56

do you think that's work?

Answer 57

13 is not an integer mod 7, though. I encourage you to think about matrices instead.

Answer 58

how with matrix

Answer 59

if A and B are nonzero matrices, is it possible that
\[A \otimes B = 0\]
where 0 is obviously the zero matrix?

Answer 60

no

Answer 61

What about these matrices
\[\left[\begin{matrix}1 &1 \\ 1 & 1\end{matrix}\right]\cdot \left[\begin{matrix}1& 1\\-1 & -1\end{matrix}\right]\]

Answer 62

yes that is right

Answer 63

What if I changed the second matrix to\[\left[\begin{matrix}1 & 2 \\ -1 & -2\end{matrix}\right]\]

Answer 64

Would that still be zero?

Answer 65

yeeeees still zero

Answer 66

So we've found matrices such that
\[A\otimes B = A\otimes C\]
and A,B,and C are all nonzero.

Answer 67

is it include which group the matrices

Answer 68

I don't know what that means.

Answer 69

i mean in this question say what is the number you choose have to be ring

Answer 70

The set of all 2x2 matrices is a group, whose composition law is element-wise addition. When you impose the additional operation of matrix multiplication, which distributes over the element-wise addition, you form a ring.

Answer 71

oh okay i understood

Answer 72

thank you so much for your cooperation. i appreciate that

Answer 73

No problem.