Can someone please help with the following problem taken from Beachy and Blair: Abstract Algebra ch 5 Commutative Rings

Give the multiplication table for the ring $\displaystyle Z_3 [x] / <x^2 - 1>$

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- Dec 17th 2012, 12:13 AMBernhardFactor Rings
Can someone please help with the following problem taken from Beachy and Blair: Abstract Algebra ch 5 Commutative Rings

Give the multiplication table for the ring $\displaystyle Z_3 [x] / <x^2 - 1>$ - Dec 17th 2012, 02:12 AMDevenoRe: Factor Rings
note that the elements of Z

_{3}[x]/<x^{2}- 1> will be cosets that we can represent as polynomials of degree < 2, to save on notation instead of writing p(x) + I, i will simply write p(u), and thus u = x + I, and i will write a instead of a + I

(taking advantage of the ring isomorphism Z_{3}-->Z_{3}[x]/<x^{2}- 1> given by: a--> a + <x^{2}+1>).

there are exactly NINE of these cosets:

0,1,2,u,u+1,u+2,2u,2u+1,2u+2. since this is a commutative ring (since Z_{3}is commutative) i will only write "half" the products (a little more than half, actually).

0*0 = 0

0*1 = 0

0*2 = 0

0*u = 0

0*(u+1) = 0

0*(u+2) = 0

0*(2u) = 0

0*(2u+1) = 0

0*(2u+2) = 0

1*1 = 1

1*2 = 2

1*u = u

1*(u+1) = u+1

1*(u+2) = u+2

1*(2u) = 2u

1*(2u+1) = 2u+1

1*(2u+2) = 2u+2 so far, these are all trivial

2*2 = 1 (4 = 1 (mod 3))

2*u = 2u

2*(u+1) = 2u+2 (remember distributivity still works)

2*(u+2) = 2u+1

2*(2u) = u (associativity still applies, as well)

2*(2u+1) = u+2

2*(2u+2) = u+1

u*u = u^{2}= (u^{2}- 1) + 1 = 1 (since u = x + I, we are essentially saying x^{2}- 1 + I = I, so x^{2}+ I = 1 + I, that is u^{2}= 1. u is a "third square root of 1". note this can't happen in a field)

u*(u+1) = u^{2}+ u = 1 + u = u+1

u*(u+2) = 2u+1 (what did i do here?)

u*(2u) = 2

u*(2u+1) = u+2

u*(2u+2) = 2u+2

(u+1)*(u+1) = 2u+2

(u+1)*(u+2) = 0 <---a zero divisor. interesting....

(u+1)*(2u) = 2u+2 <---note that cancellation does NOT hold (u+1)*(u+1) = (u+1)*(2u), but u+1 does not equal 2u.

(u+1)*(2u+1) = 0 <---more zero divisors.

(u+1)*(2u+2) = u+1...but u+1 is NOT the multiplicative identity. very bizarre, hmm?

(u+2)*(u+2) = u+2 <--more strangeness

(u+2)*(2u) = u+2

(u+2)*(2u+1) = 2u+1 <---u+2 behaves VERY strangely. note that 1 is a root of x^{2}-1, and so is u, so u+2 is "sort of like" 1+2 = 0.

(u+2)*(2u+2) = 0

(2u)*(2u) = u

(2u)*(2u+1) = 2u+1

(2u)*(2u+2) = u+1

(2u+1)*(2u+1) = u+2

(2u+1)*(2u+2) = 0

(2u+2)*(2u+2) = 2u+2

all in all, a very strange ring of order 9. - Dec 17th 2012, 02:15 AMBernhardRe: Factor Rings
Will work through this now

Peter - Dec 17th 2012, 02:27 PMDrexel28Re: Factor Rings
I assume you know the Chinese Remainder Theorem, so that $\displaystyle \mathbb{Z}_3[x]/(x^2-1)\cong \mathbb{Z}_3[x]/(x-1)\times\mathbb{Z}_3[x]/(x+1)\cong\mathbb{Z}_3^2$.

- Dec 17th 2012, 03:21 PMDevenoRe: Factor Rings
that's true, alex, but it's not the "usual" (field) multiplication on Z

_{3}xZ_{3}.

but let's use this:

say we want to multiply (u+1)(u+2).

first, which element of Z_{3}xZ_{3}does u correspond to? a little thought shows it must be (1,2) (a root of x-1 in the first factor, and a root of x+1 in the second factor).

since we want to preserve the identity, we have to send 1-->(1,1), the identity of Z_{3}xZ_{3}.

so u+1 = (1,2) + (1,1) = (2,0) <---this explains why u+1 is a 0-divisor.

similarly, u+2 = (1,2) + (2,2) = (0,1) <--explains why u+2 is a 0-divisor.

so (u+1)(u+2) = (2,0)(0,1) = (0,0) = 0

continuing, we can now create a complete correspondence between the 2 versions:

0<-->(0,0) (duh!)

1<-->(1,1)

2<-->(2,2)

u<-->(2,1)

u+1<-->(0,2)

u+2<-->(1,0)

2u<-->(1,2)

2u+1<-->(2,0) wait...was 2u+1 a 0-divisor? yep, sure was.

2u+2 <-->(0,1)

proving this IS an isomorphism is tedious (additively it's clear, but multiplication is a chore) perhaps the distributive law might help.... - Dec 17th 2012, 03:27 PMDrexel28Re: Factor Rings
No, it's the multiplication given by the product ring structure, which is not that bad.

- Dec 17th 2012, 08:10 PMDevenoRe: Factor Rings
i think this is a great example of how the additive structure of a ring can have very little to do with the multiplicative structure. and that different multiplicative structures on the very same abelian group can produce VERY different results. put another way: the cardinality of a finite ring tells us a lot less than we might hope.