Example of Subring is commutative whereas whole ring is not commutative??

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- Nov 11th 2012, 10:05 PMpuneethaRings
Example of Subring is commutative whereas whole ring is not commutative??

- Nov 11th 2012, 10:10 PMpuneethaRe: Rings
help me out plllllllllll

- Nov 11th 2012, 11:33 PMDevenoRe: Rings
sure.

let R = the ring of all 2x2 matrices with integer entries. this is clearly a non-commutative ring (for example, the matrices:

$\displaystyle A = \begin{bmatrix}1&0\\1&0 \end{bmatrix};\ B = \begin{bmatrix}1&1\\0&0 \end{bmatrix}$

do not commute).

but the matrices of the form:

$\displaystyle K = \begin{bmatrix}k&0\\0&k \end{bmatrix},\ k \in \mathbb{Z}$

form a sub-ring of R isomorphic to the integers, which is thus commutative. - Nov 11th 2012, 11:53 PMpuneethaRe: Rings
thank u sooooooo muchhhhhhhhh :) :)