Assumptions in group, ring and field theory

In group ring and field theory we often assume that we can add the same thing to both sides of an equation or multiply both sides of an equation by the same element.

How do we formally justify these operations or are they axioms?

That is, what is the rigorous and formal proof of the following:

1. If a = b then a+c = b + c and c + a = c + b

2. If a = b then ca = cb and ac = bc

Would appreciate help in this matter.

Peter

Re: Assumptions in group, ring and field theory

the ancient term for such things is "al-muqabala" which roughly translates as "balancing" (this is from the title of the famous work by Al-Khwarizmi called: al-Kitab al-mukhtasar fi hisab al-jabr w'al-muqabala , or: the compendious book on computation by completion and balancing, the term "al-jabr" refers to transforming a + c = b to a = b - c, from which we derive our term "algebra").

note that a binary operation on a set S, such as + or * is defined as a FUNCTION (a special kind of relation R for which xRy and xRz implies y = z) from SxS to S.

so (for example) if +(a,b) = +(a,c), because + is a function we have: (a,b) = (a,c) in SxS. but equality in SxS is defined as: (x,y) = (u,v) if and only if x = u, and y = v.

applying this to x = a, y = b, u = a, v = c, we get: a = a (well...ok....) and b = c (ding!).

one can also see +c (for example) as a function from S to S: +c(a) = a+c (this kind of thing is called a "right translation", which is the same thing as a "left translation" for commutative operations, but the two ideas differ for non-commutative operations. for subgroups, translates are better known as their "other" name, cosets). since these are FUNCTIONS, it follows automatically that if a = b, +c(a) = +c(b) (functions can only have ONE image for identical domain elements).

the common phrase to describe all this is: "equals to equals are equal" if i have two things, which are the same, and i do the same thing to each one, i get the same result:

a = b (so we can use b in place of a, and vice versa, because they are EQUAL)

a+c = b+c

this illustrates the "one-way-ness" of functions, from a = b, we always have f(a) = f(b) for ANY function f (whether f "adds c" or "multiplies by c" or many other things you can imagine), we often CANNOT go the other way (not all functions are reversible).

Re: Assumptions in group, ring and field theory

Thanks for that ... most interesting ...

*Essentially it seems that you are saying that the reason 'balancing' processes are valid is because of the definition of binary operations.*

That is specifically the reason a + c = b + c follows from a = b because of the definition of the binary operation or function +c ... and then a similar argument would apply to *c

I have not seen this explained in the algebra texts I have been using (but maybe I have not looked closely enough!) ... will go back and look again ... I must say up to now adding a constant to both sides of an equation and multiplying an equation through by an element seemed intuitively valid ... and that bothered me since I was not sure of the reason for it being valid.

Peter