. Let R be a ring and let x € R. We inductively define¹ multiplication of elements of R by ositive integers: for x ER, we set 1x = x (here 1 = 1z is the multiplicative identity of Z), nd for ne N, we set (n+1)x= nx+x. (a) Prove that (m+n)x= mx +nx for all m, n € N and x € R. (b) Prove that n(x + y) = nx+ny for all n € N and x, y € R. (c) Prove that (mn)x= m(nx) for all m, n € N and x € R. ote that it is not the case that these properties are special cases of the distributive and ssociative properties in the ring R, because m and n are positive integers, and cannot be ssumed to be elements of R, an arbitrary ring. With the notation of the preceding problem, we have 2r= (1+1)r= lr+r=r+r
. Let R be a ring and let x € R. We inductively define¹ multiplication of elements of R by ositive integers: for x ER, we set 1x = x (here 1 = 1z is the multiplicative identity of Z), nd for ne N, we set (n+1)x= nx+x. (a) Prove that (m+n)x= mx +nx for all m, n € N and x € R. (b) Prove that n(x + y) = nx+ny for all n € N and x, y € R. (c) Prove that (mn)x= m(nx) for all m, n € N and x € R. ote that it is not the case that these properties are special cases of the distributive and ssociative properties in the ring R, because m and n are positive integers, and cannot be ssumed to be elements of R, an arbitrary ring. With the notation of the preceding problem, we have 2r= (1+1)r= lr+r=r+r
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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1
Transcribed Image Text:1. Let R be a ring and let x € R. We inductively define multiplication of elements of R by
positive integers: for x ER, we set 1x = x (here 1 = 1z is the multiplicative identity of Z),
and for n € N, we set (n+1)x= nx+x.
=
(a) Prove that (m + n)x = mx + nx for all m, n E N and x € R.
(b) Prove that n(x + y) = nx + ny for all n N and x, y € R.
(c) Prove that (mn)x= m(nx) for all m, n E N and x € R.
Note that it is not the case that these properties are special cases of the distributive and
associative properties in the ring R, because m and n are positive integers, and cannot be
assumed to be elements of R, an arbitrary ring.
With the notation of the preceding problem, we have
Similarly, we have
2x = (1+1)x= 1x + x = x+x.
3x =
= (2+1)x = 2x + x = (x+x) + x = x+x+x.
And so on. This is why, for n € N, the element nx is sometimes described informally as
the element of R obtained by adding n summands, all equal to x; but this description is
just that: informal. Such a description is not sufficiently rigorous to establish the important
properties of the multiplication operation of N on R.
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