Rings and Subrings - Dummit and Foote Chapter 7

I am reading Dummit and Foote on Ring Theory - ch 7

On page 228 D&F give examples of subrings

For example 2 they write

"$\displaystyle 2 \mathbb{Z} $ is a sub-ring of $\displaystyle \mathbb{Z} $ as is $\displaystyle n \mathbb{Z} $ for any integer n.

The ring $\displaystyle \mathbb{Z} / n \mathbb{Z} $ is not a subring of $\displaystyle \mathbb{Z} $ for any $\displaystyle n \geq 2 $ "

My question is surely the ring $\displaystyle \mathbb{Z} / n \mathbb{Z} $ **is ** a subring of $\displaystyle \mathbb{Z} $ if we identify the equivalence class $\displaystyle \overline{0}$ with 0 and $\displaystyle \overline{1} $ with 1

Perhaps the point D&F are making is that 0 and 1 are not the same as $\displaystyle \overline{0}$ and $\displaystyle \overline{1} $?

Can someone help by clarifying this issue

Peter

Re: Rings and Subrings - Dummit and Foote Chapter 7

no.

Z/nZ is finite (in fact, it has just n elements), whereas any non-trivial subring of Z is infinite.

to see this: note that since the integers are well-ordered, if a subring S of Z has any element x besides 0, then one of {x,-x} is positive. since S contains some positive integers, it must contain a LEAST positive integer, call it y.

by closure, S must also contain y+y,y+y+y,y+y+y+y,.... in short, S must contain every positive integer of the form ny, where n is a natural number. since there are infinitely many natural numbers n, there are infinitely many elements of S. you see, for two INTEGERS my and ny, if my = ny, then m = n.

this is NOT true for Z/nZ. for example:

4*3 = 0 (mod 6)

2*3 = 0 (mod 6)

but 4 ≠ 2 (mod 6).

it is very perilous to consider 1 (mod n) as "the integer 1". one should actually think of 1 (mod n) as the SET (a coset, actually):

1 + nZ = {......,-4n+1,-3n+1,-2n+1,-n+1,1,n+1,2n+1,3n+1,4n+1,.....}

any element of which "represents" the set (so if we are working mod 6, 5 and 11 represent "the same element of Z/6Z").

the representatives are integers, but what they represent are NOT integers. the senator from Wyoming represents Wyoming (as does any senator from Wyoming), but said senator is NOT the state, but just a "one-person stand-in" (it would clearly be impractical to house the entire population of Wyoming in the House of Congress just for the purpose of voting on legislation).

loosely speaking, what we get in Z/nZ is similar to what happens when we turn the unit interval [0,1] into a circle via a map t → (cos(2πt),sin(2πt)). in that case, we declare 0 = 1. this map can be extended to the entire real line, by "wrapping around infinitely many times".

in Z/nZ, we effectively say: n = 0, but all the other arithmetic of the integers stays the same. this turns the "line" Z into a loop with n nodes (like a clock). there is a fundamental difference between Z and Z/nZ: all of the elements of Z/nZ have finite additive order, only 0 has finite additive order in Z.

QUOTIENT structures (those obtained by "identifying" all the elements of a distinguished sub-structure) are quite different than sub-structures (which behave more or less like their "parents" do).

Re: Rings and Subrings - Dummit and Foote Chapter 7

Thank you, your post was extremely helpful!

Peter