Subrings

Xxhaylsxx

Hi probably an insanely stupid question but just cant get my head around this at all. How to you show that a subset is a subring of say R or C. I know you must show that it has the multiplicative unit, that it is an additive subgroup and that x,y is element of B and xy is an element of B. But when it comes to specific examples such as: let p be prime. Define Zp = (m/nis element of Q). show that Zp is a subring of Q, I get totally confused. Anybody got any guidance : (
Thanks

Fremen

Well, first show each element of the subring is contained in the ring, then show the ring axioms hold in the subring one by one. It's exactly the same method you'd prove something is a ring
Your definition of Z_p seems a bit crazy, are you sure you're not mixing up two different rings? Z_p (set of integers modulo p) isn't a subring of Q since the multiplication operations are different.

Edit:
See if you can prove that the set of NXN matrices with real entries form a ring, and that the set of NXN invertible matrices form a subring of that ring.
Post any problems you have here.

Fremen

Fremen wrote: »

Well, first show each element of the subring is contained in the ring, then show the ring axioms hold in the subring one by one. It's exactly the same method you'd prove something is a ring
Your definition of Z_p seems a bit crazy, are you sure you're not mixing up two different rings? Z_p (set of integers modulo p) isn't a subring of Q since the multiplication operations are different.

Edit:
See if you can prove that the set of NXN matrices with real entries form a ring, and that the set of NXN invertible matrices form a subring of that ring.
Post any problems you have here.

Sorry, my bad, they don't form a subring. Invertibility isn't conserved by matrix addition. I was pretty tired when I wrote that.

seandoiler

maybe in your tired state you confused invertible and diagonal matrices as being a suitable subring

dabh

Xxhaylsxx wrote: »

Hi probably an insanely stupid question but just cant get my head around this at all. How to you show that a subset is a subring of say R or C. I know you must show that it has the multiplicative unit, that it is an additive subgroup and that x,y is element of B and xy is an element of B. But when it comes to specific examples such as: let p be prime. Define Zp = (m/nis element of Q). show that Zp is a subring of Q, I get totally confused. Anybody got any guidance : (
Thanks

Check out one-by-one the conditions in the definition of subring. Does zero belong to the subset? Does the sum of any two elements of the set belong to the set, etc.?

OP needs to define Zp more clearly. If this were simply the set of integer multiples of the prime number p it would be closed under addition and multiplication but would not contain a multiplicative identity element.

But maybe 'Zp' could be the set of all rational numbers that can be expressed in the form m/n, where n is not divisible by p. If this is what the OP means, it makes the problem more interesting. Show that the sum of two rational numbers expressible in this form must itself be expressible in this form etc.