Define a+b=a+b -1 and a*b=ab-(a+b)+2 Assume that (Z, +,*) is a
ring. (a) Prove that the...
Define a+b=a+b -1 and a*b=ab-(a+b)+2 Assume that (Z, +,*) is a
ring. (a) Prove that the additative identity is 1? (b) what is the
multipicative identity? (Make sure you proe that your claim is
true). (c) Prove that the ring is commutative. (d) Prove that the
ring is an integral domain. (Abstrat Algebra)
Consider the ring R = Z ∞ = {(a1, a2, a3, · · ·) : ai...
Consider the ring R = Z ∞ = {(a1, a2, a3, · · ·) : ai ∈ Z for
all i}. It turns out that R forms a ring under the operations (a1,
a2, a3, · · ·) + (b1, b2, b3, · · ·) = (a1 + b1, a2 + b2, a3 + b3,
· · ·), (a1, a2, a3, · · ·) · (b1, b2, b3, · · ·) = (a1 · b1, a2 ·
b2, a3 ·...
Consider the ring R = Z∞ = {(a1,a2,a3,···) : ai ∈ Z for all
i}.
It...
Consider the ring R = Z∞ = {(a1,a2,a3,···) : ai ∈ Z for all
i}.
It turns out that R forms a ring under the operations:
(a1,a2,a3,···)+(b1,b2,b3,···)=(a1 +b1,a2 +b2,a3 +b3,···),
(a1,a2,a3,···)·(b1,b2,b3,···)=(a1 ·b1,a2 ·b2,a3 ·b3,···)
Let I = {(a1,a2,a3,···) ∈ Z∞ : all but finitely many ai are 0}.
You may use without proof the fact that I forms an ideal of R.
a) Is I principal in R? Prove your claim.
b) Is I prime in R? Prove your claim....
Consider the ring R = Z∞ = {(a1,a2,a3,···) : ai ∈ Z for all
i}.
It...
Consider the ring R = Z∞ = {(a1,a2,a3,···) : ai ∈ Z for all
i}.
It turns out that R forms a ring under the operations:
(a1,a2,a3,···)+(b1,b2,b3,···)=(a1 +b1,a2 +b2,a3 +b3,···),
(a1,a2,a3,···)·(b1,b2,b3,···)=(a1 ·b1,a2 ·b2,a3 ·b3,···)
Let I = {(a1,a2,a3,···) ∈ Z∞ : all but finitely many ai are 0}.
You may use without proof the fact that I forms an ideal of R.
a) Is I principal in R? Prove your claim.
b) Is I prime in R? Prove your claim....
Answer b please...
Let R be a ring and let Z(R) := {z ∈ R :...
Answer b please...
Let R be a ring and let Z(R) := {z ∈ R : zr = rz for all r ∈
R}.
(a) Show that Z(R) ≤ R. It is called the centre of R.
(b) Let R be the quaternions H = {a+bi+cj+dk : a,b,c,d ∈ R} and
let S = {a + bi ∈ H}. Show that S is a commutative subring of H,
but there are elements in H that do not commute with elements...