Why are groups, rings, and fields defined in the way that they are?

[gravenewworld]

Groups, Rings, Fields?

I know what groups, rings, and fields are. My question is why are groups, fields, and rings defined the way they are? Why did mathematicians chose the properties that they did that define groups, rings, and fields? What is so special about those properties? Why couldn't they have chosen completely different other properties in order to define groups etc?

 

[mathwonk]

i think to answer this you would have to ask yourself why groups, rings, and fields, (defined the way they are), are important. It was the importance of these collections of rpoeprties that caused them to be codified with special names.

Rings basically generalize the integers, the polynomials, and the matrices, all pretty important.

groups probably arose as permutations in galois' analysis of solutions of equations, but also arise as the units in a ring, and as families of invertible mappings, another ubiquitous concept.

fields occur when we enlarge any (commutative) ring to allow divison, as in forming the rationals from the integers. field extensions also arise when trying to construct solutions of polynomials.

 

[matt grime]

In my view, you're looking at it the wrong way. We didn't suddenly decide to study sets with two binary operations satisfying some axioms plucked at random from thin air thinking that field was a word that needed a meaning in maths. We study objects, decide what the important features are and see if by purely considering an abstract object with those features if we can get any good general theorems. The complex, real and rational numbers all have similar algebraic properties (in the sense of addition and multiplication) - we pick them out and see what we can say about these, and if there are any other objects with such formal properties.

I think mathwonk needs to add "(commutative) ring with no zero divisors" to his final paragraph, or "to allow division of the elements that do not divide zero".

 

[mathwonk]

matt grime is explaining which are the elements of a ring whose reciprocals can be added to a ring, without causing any elements of the ring to become zero.

i.e. if we add in the reciprocal of x, then for any element y such that xy = 0, we will have y = y1 = y(x)(1/x) = 0. so y will become zero.

since zero is not supposed to equal one, in this case we cannot add in both the reciprocals of x and of y, since that would cause 1 = x(1/x)y(1/y) = xy(1/x)(1/y) = 0.


but in the larger sense, both matt and i are trying to discuss the question of why are certain definitions made, which we believe to have been your actual question.

simply put (by my algebra teacher): if a concept is important enough, we make it a definition so we will recognize when we see it again.