Prove that (1)×x = x in every ring. (N.B. This question is about rings in genera

Question

Prove that (1)×x = x in every ring. (N.B. This question is about rings in general, not just the integers. So you can only use R1 through R8 when answering this question.)

R1 Addition is associative.

R2 Addition is commutative.

R3 There is an element 0 such that x + 0 = x for all elements x.

R4 For every element x there is an element y such that x + y = 0.

R5 Multiplication is associative.

R6 Multiplication is commutative.

R7 There is an element 1 such that x × 1 = x for all elements x.

R8 Multiplication distributes over addition.

Explanation / Answer

x=1.x by R7

1.x+(-1).x =(1-1).x, by R8

0.x

Now write 0=(0+0)

0.x=(0+0).x

0.x=0.x+0.x,R8

0=0.x+0.x-0.x

=0.x

That is 0.x=0

=0.x+0.x

Hence 1.x+(-1).x=0

(-1).x is multiplicative inverse of 1.x

And 1.x=x by R7

(-1).x=-x,for every x in ring

Related Questions

Navigate

• Browse (All)
• Browse P
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.