From: kovarik@mcmail.cis.McMaster.CA (Zdislav V. Kovarik) Subject: Re: Functions on domains as rings with unity? Date: 12 Jul 2000 04:20:17 -0400 Newsgroups: sci.math Summary: [missing] My reply is long; you might be better off exploiting an instructor or a TA, who are paid to do the explaining. But if it is self-study, bear with me. In article <8kh2mt$u87$1@nnrp1.deja.com>, wrote: :I am finding myself unable to go about solving this problem in any :satisfactory way. First, the problem : :----------- :Are the following sets of functions commutative rings with unity? :(a) all functions f on a domain D for which f(0)=0 :(b) all functions f on D with f(0)=f(1) :(c) all functions on D with f(0)!=0 :(d) all functions f on Q (the rational field) with -7<=f(x)<=7 for all x :(e) all f on Q with f(x+1)=f(x) for all x (such an f is periodic). :----------- : :As best as I can tell the approach for a-e is to either prove the axioms :for commutative rings hold or find counterexamples that contradict the :axioms. : :I think I have two major points of confusion: : : (1) Precisely how the ring axioms are to apply to function objects :which seem quite different from the usual "constant" elements. : The first useful fact is that the set of all functions on D is a commutative ring with identity, provided (what the textbook either said right there or assumed tacitly) that: the functions have numbers as values (we will pass on the more general situations), and the addition and multiplication is performed pointwise, that is, (f+g)(x) = f(x) + g(x) for all x in D, (f*g)(x) = f(x) * g(x) for all x in D. The identity is the constant function with value 1. (Scan the axioms now, and see that they hold.) What the problems (a) to (e) want from you is checking (A) if the given set "survives" addition and multiplication (if the sum and the product of two functions from that set is again in that set, etc.) (B) if it has its own unit (which may or may not be the unit I mentioned). Things may get clearer if you look up the concept "subring", and a little theorem about how to characterize subrings (the core of the theorem is that it spares you of the necessity to scan all axioms, and allows you to concentrate on the ones that are not automatically satisfied.) :(2) My mind just seems blank when thinking about how to precisely :reason with these objects. Like what to look out for and think about, :etc. I don't really know how to express what I am trying to say here. I tried to give a tip in the previous paragraph. The main mental trick is to get used to treating functions as if they were numbers (adding them and multiplying them as indicated. Example: for f(x) = 1+x and g(x) = 2+3*x, you get (f+g)(x) = 3+4*x and (f*g)(x) = 2+5*x+3*x^2.) ***(Remark that can be skipped and read later:) Some caution has to be exercised: for example, the axioms of a ring do not promise division by nonzero elements, so division is not always possible. Also, the axioms do not even mention ordering, so we cannot expect the same kind of ordering in the ring (or any kind, if the ring is more unlike the real numbers.) ***(End of remark) :If the text had included some worked examples or if some had been done :in class then perhaps I wouldn't be so confused. You have a valid point here. Look around for textbooks which do have worked examples. Perhaps Schaum's Series? :(My intuition is that :(a) is a commutative ring, but I don't know how to satisfactorily :explain this.) Let me work this one out for you. It says: :(a) all functions f on a domain D for which f(0)=0 (It is tacitly assumed that 0 is in the domain.) Does the set withstand addition and multiplication? Given f and g in the set, we are told that f(0)=0 and g(0)=0. Give the sum f+g a name, say h, and give the product f*g a name, say k: then for all x in D, h(x) = f(x) + g(x) and k(x) = f(x) * g(x) Find out what h(0) and k(0) are: h(0) = f(0) + g(0) = 0 + 0 = 0, similarly k(0) = 0. So, f+g and f*g are also in the set, and we have a subring of a commutative ring, hence a commutative ring in its own right. The unit is trickier (and is a shock for the untrained): the constant function with value 1 is not in the subring! But let's try for a substitute: all you need is that it leaves functions unchanged by multiplication. So, the function e: e(x) = 1 for all x other than 0, but e(0)=0 does just that: we invented a unit for the smaller ring. Check: e(x)*f(x) = 1 * f(x) = f(x) for x other than 0, e(0)*f(0) = 0 * f(0) = 0*0 = 0 = f(0) So, e*f = f for all f in the smaller ring defined in (a). Your intuition was right. For the other sets, just brief hints: :(b) all functions f on D with f(0)=f(1) similar to (a), even easier :(c) all functions on D with f(0)!=0 does it contain f-f for every f? :(d) all functions f on Q (the rational field) with -7<=f(x)<=7 for all x does it contain the function 7*1 + 7*1? :(e) all f on Q with f(x+1)=f(x) for all x (such an f is periodic). as easy as (b) Good luck, ZVK(Slavek).