Let r be a communtative ring and a, b elements in r. prove that if a and b are units, then so is ab. what can we say about ab when a is a unit and b is a zero divisor? prove your claim.

Answer with  explanation:

Let R be a communtative ring .

a and b elements in R.Let a and b are units

1.To prove that ab is also unit in R.

Proof: a and b  are units.Therefore,there exist elements u and v such that

au=1 and bv=1 ( by definition of unit )

Where u and v are inverse element  of a and b.

(ab)(uv)=(ba)(uv)=b(au)(v)=bv=1 ( because ring is commutative)

Because bv=1 and au=1

Hence, uv is an inverse element of ab.Therefore, ab is a unit .

Hence, proved.

2. Let a is a unit and b is a zero divisor .

a is a unit then there exist an element u

such that au=1

By definition of unit

b is a zero divisor then there exist an element

such that bv=0 where

By definition of zero divisor

(ab)(uv)=b(au)v    ( because ring is commutative)

(ab)(uv)=b.1.v=bv=0

Hence, ab is a zero divisor.

If a is unit and b is a zero divisor then ab is a zero divisor.

y = 3x+4

step-by-step explanation:

to find the slope

slope = (y2-y1)/ (x2-x1)

    = (-8-4)/ (-4-0)

    = -12/-4

  = 3

using the slope intercept form of a line

y= mx+b

we know the slope is 3

we know the y intercept is 4 ( the point where it crosses the y axis or x=0)

y = 3x+4