Question

(5 points each) Define a new operation of addition in Z by x Oy = x + y - 1 with a new multiplication in Z by x Oy = 1. (a) Is Z a commutative ring with respect to these operations? (b) Find the unity, if one exits.

Answer 1

Operations on Z are given by ay = x+y-1 and * Oy=1 1 clearly x 04, 20YE Z Vaiye Z ..Z is closed wet addition and multiplication Acossiativity wet wet addition: let xiy, zE Z be any Telements. onsider OC Đly #2) x [ 4+2-1] Os +(y +/-1)-2 x+y+z-2 ( 29) © Z (oty - z g+ 3-1 + 2 - - xtytz-2. cc @y@z) = (a® y®z :. Associative property is satisfied Additive identity: Let e EZ for an inverse a e= xte-1 = x riezt is an additive identity.

additive inverse let x'E Z be an element. for an inverse, we can write an expression x 6 x'=e at oc'-IEI xxx'=2 = 2-2 ::. 2-x is an additive inverse of ae. Commutative property Wet addition Consider x y y 0 x =xt y-1 = y + x-1 = 2ty-1 y Y ② x rary ez Hence Commutative property is satisfied. Associative property.coet multiplication x 402) let X01 (xoy) oz soz 2 x (4 oz) = = ( ay) Oz :. A ssociative property is satisfied:

Distributive law Consider, 20 YOZ = 20 (y+Z-1 = 1 Again XOYO ZOZ - jūs Its = ܫܕ SI :'. TO y z = (x Oy) 20Z Similarly we we can show that, (2@y) 02 = (a a02) (YOZ (9c :. Distributive laws are hold. This shows that Zis a ling. Fora commutative eing xOy = =f & yox=1 ,: xoy = yox W X1Y6Z Hence Zis a commutative eing.

CD For a unity Unity means mal ti plicative identity. So e' be the multiplicative ident lentity . xo e = x 1 = x But we could not find el. Hence unity doesit exists.