What is "Modding Out by A" in R/A?

[math-chick_41]

R/A "mod out by A"??

Hi, can someone please explain to me what "modding out by A" means?
R is ring and A is subring of R
R/A = {r+A|r in R} I understand that R/A is a bunch of left cosets
and when A is an ideal of R then R/A is ring
but what is all this talk of modding out?

any input would be great

 

[Hurkyl]

This whole thing is a generalization of modulo arithmetic. The basic idea is that the ideal is specifying the set of things that are now going to be zero (much like the "mod 5" specifies that all multiples of 5 are now equal to zero).

 

[fourier jr]

Hi, can someone please explain to me what "modding out by A" means?
R is ring and A is subring of R
R/A = {r+A|r in R} I understand that R/A is a bunch of left cosets
and when A is an ideal of R then R/A is ring
but what is all this talk of modding out?
any input would be great

"mod out by A" is just the math lingo for "form the quotient ring R/A={r+A|r in R}". & it's only defined if A is an ideal, not just a subring of R.

 

[selfAdjoint]

The elements of the modulo ring are the equivalence classes of elements equal in the modded-out structure. The classes inherit their algebraic relationships from the big ring.

You have to check that products of things from the big ring with things from the little one stay in the little one in order to do this in a well-defined way. That is, the little structure has to actually be an ideal in the big ring.

 

[mathwonk]

it is a basic idea in math of considering some thigns as equaivalent to each other, and looking at the collection of equivalence classes as forming a new object.example: consider the set S of all pairs of integers (n,m) where m is not zero, and equate two such pairs (n,m) = (a,b) iff am=bn. Then denote the class of all pairs equivalent to (n,m) as n/m. I.e. n/m = a/b iff am = bn. The set Q : (S/=) of these equivalence classes is called the set of rational numbers.or consider the set S of all Cauchy sequences {a(n)} of rational numbers, where two sequences {a(n)} = {b(n)} are equivalent iff the difference sequence {a(n)-b(n)} converges to zero. The set R:(S/=) is called the set of real numbers.

Notice you can add and multiply Cauchy sequences, and they form a ring, and the sequences which converge to zero is an ideal in that ring, so we are getting the reals by "modding out" the ring S of Cauchy sequences by the ideal of "null sequences".

notice if a Cauchy sequence does not converge to zero, then its elements are eventually non zero and we can form a Cauchy sequence eventually consisting of their inverses, hence the product sequence is eventually all 1's, hence differs from the dientity sequence (all 1's) by a null sequence.

Thus the ideal of null, sequences is a maximal ideal, and hence the result of modding out by it is a field, the field R of real numbers.If we define two integers to be equivalent if their difference is divisible by some positive integer n, we get an equivalence realtion whose set of equivalence classes is the ring Z(n) of "integers mod n". The ideal we modded Z out by is the ideal of all multiples of n.Bertrand Russell tried to define cardinal numbers by considering the set of all finite sets, and equating two sets if there is a bijection between them. Then the collection of equaivalence classes he called the set of finite cardinal numbers. But there are some logical problems with considering such a large collection to begin with, so other people tend to choose a particular collection of sets to represent the cardinal numbers, like their fingers and toes.there is literally no end to the use of this construction. In general however most people find it sdifficult to think of a big equivalence class of objects as one object so they tend to look for one special object in each class to represent the whole class.

E.g. in Z(n) it is normal to choose "least residues" to represent the classes of modular integers, as Gauss did,

i.e. then we write Z(n) = {[0], [1], ...[n-1]}.In mlinear algebra we often equate two matrices if they are "similar" i.e. A, B are similar if B = SAS^(-1) where S is invertible.

then we look for special representatives in each class, like rational canonical forms, or jordan normal forms, etc...in topology we define equivalence of apths by homotopy, and define the set of calsses as the first homotopy group, or fundamental group.

for the circle, the group is siomorphic to Z, and usually we pick the representative path t-->(cos(2pi.nt), sin(2pi.nt)) to represent the class of n.

on and on and on...

indeed all mathematical construictions are of this type. You find an interesting set, and to study it single out an interesting property, and equate things which resemble each other with respect to that property.In a ufd we define units as invertible elements, and then define the relation of "associate" between two prime elements to mean that one is a unit times the other. then the set of associate classes of primes, together with the units, represents the set of all primes, and if we choose one prime from each class, we can express any element uniquely as a product of (powers of) a finite set of those primes and one unit.