Math 476 - Abstract Algebra 1
Day 28 Group Assignment Name:
Integer Multiples and Exponents
Definition 8.2 Let Rbe a ring, let n≥3 be an integer, an let x1, x2,··· , xn∈R. Then we define x1+x2+···+xn=
(x1+x2+···+xn−1) + xnand x1x2· · · xn= (x1x2···xn−1)xn.
Note: Definition 8.2 is an example of a recursive definition. That is, it defines the next instance in terms of one or more
previous instances. To complete a particular computation, one would need to apply the recursive definition multiple times.
For example, to compute x1+x2+x3+x4, we could apply the definition as follows:
x1+x2+x3+x4= (x1+x2+x3) + x4;x1+x2+x3= (x1+x2) + x3; compute x1+x2; use the result to compute
(x1+x2) + x3=x1+x2+x3. Then use this second result to compute (x1+x2+x3) + x4=x1+x2+x3+x4.
Definition 8.3 Let Rbe a ring, and let x∈R. The the expression 1xand x1are both defined to be equal to x; that is,
1x=xand x1=x.
Furthermore, for every integer n≥2, we define the expressions nx and xnrecursively as follows:
•nx =x+x+···+x
|{z }
nterms
=x+x+· · · +x
|{z }
n−1 terms
+x= (n−1)x+x.
•xn=x·x···x
|{z }
nfactors
= (x·x···x)
|{z }
n−1 factors
x=xn−1x.
Note: We would like to extend these ideas to subtraction, to negative exponents, and to be able to use zero as an exponent.
As we will soon see, some of these notions will work for arbitrary elements in any ring, but others will require our ring and/or
element to have additional properties. For example, we can only take x0= 1 if our ring has a multiplicative identity. Also,
x−n=x−1nonly makes sense when xis a unit (otherwise, x−1does not exist!).
Definition 8.4 Let Rbe a ring, and let nbe a positive integer.
•For all x∈R, we define 0x= 0Rand (−n)x=n(−x), where −xis the additive inverse of x.
•If Ris a ring with identity, then for each nonzero x∈R, we define x0= 1R. If Rdoes not have identity, then x0
remains undefined.
•If Ris a ring with identity, then for each unit x∈R, we define x−n= (x−1)n, where x−1denotes the multiplicative
inverse of x. If Rdoes not have identity, or if xis not a unit in R, then x−nremains undefined.
Note: If nis a negative integer, then Definition 8.4 implies that nx = (−n)(−x) and xn=x−1−n, where −n > 0. These
methods of expressing elements will be useful when proving later results in this section.
Theorem 8.5 Let Rbe a ring and let x, y ∈R, and let mand nbe integers. Then
i). m(x+y) = mx +my. ii). −(mx) = m(−x) = (−m)x
iii). (m+n)x=mx +nx iv). m(nx) = (mn)x
v). m(xy) = (mx)y=x(my) vi). (mx)(ny) = mn(xy).
1. The goal for this activity is to outline a proof for Theorem 8.5 part i).
(a) Briefly explain why parts i) and iii) of Theorem 8.5 are not an immediate consequence of our Ring Axioms.
(b) Our strategy for proving Theorem 8.5 part i) will be to consider three cases: m= 0, m > 0, and m < 0. Prove
the first case, when m= 0.
