Let R be a ring and n ∈ N. Let S = Mn(R) be the ring of n × n
matrices with entries in R.
a) i) Let T be the subset of S consisting of the n × n diagonal
matrices with entries in R (so that T consists of the matrices in S
whose entries off the leading diagonal are zero). Show that T is a
subring of S. We denote the ring T by Dn(R).
ii). Show...
If R is the 2×2 matrices over the real, show that R has
nontrivial left and right ideals.?
hello
could you please solve this problem with the clear hands writing to
read it please? Also the good explanation to understand the
solution is by step by step please
thank
the subject is Modern
Algebra
Let R be a ring (not necessarily commutative), and let X denote
the set of two-sided ideals
of R.
(i) Show that X is a poset with respect to to set-theoretic
inclusion, ⊂.
(ii) Show that with respect to the operations I ∩ J and I + J
(candidates for meet and join;
remember that I+J consists of the set of sums, {i + j} where i ∈
I and j ∈ J) respectively,
X is a lattice.
(iii) Give...
10.3.6 Exercise: Product of Pairwise Comaximal Ideals. Let R be
a commutative ring, and let {A1,...,An} be a pairwise comaximal set
ofn ideals. Prove that A1 ···An = A1 ∩ ··· ∩ An. (Hint: recall that
A1 ···An ⊆ A1 ∩···∩An from 8.3.8).
6.4.13. If R is the ring of Gaussian integers, show that Q(R) is
isomorphic
to the subfield of C consisting of complex numbers with rational
real and
imaginary parts.
In this exercise, we will prove the Division Algorithm for
polynomials. Let R[x] be the ring of polynomials with real
coefficients. For the purposes of this exercise, extend the
definition of degree by deg(0) = −1. The statement to be proved is:
Let f(x),g(x) ∈ R[x][x] be polynomials with g(x) ? 0. Then there
exist unique polynomials q(x) and r(x) such that
f (x) = g(x)q(x) + r(x) and deg(r(x)) < deg(g(x)).
Fix general f (x) and g(x).
(a) Let...