c) Let R be any ring and let ??(?) be the set of all n by...
c) Let R be any ring and let ??(?) be the set of all n by n
matrices. Show that ??(?) is a ring with identity under standard
rules for adding and multiplying matrices. Under what conditions is
??(?) commutative?
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...
let R be a ring; X a non-empty set and (F(X, R), +, *) the ring
of the functions from X to R. Show directly the associativity of
the multiplication of F(X, R). Assume that R is unital and
commutative. show that F(X, R) is also unital and commutative.
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...
Let R be a ring and f : M −→ N a morphism of left R-modules. Show that:
c) K := {m ∈ M | f(m) = 0} satisfies the Universal Property of Kernels.
d) N/f(M) satisfies the Universal Property of Cokernels.
Q2. Show that ZQ :a) contains no minimal Z-submodule
Let R be a ring and f : M −→ N a morphism of left R-modules. Show that:
c) K := {m ∈ M | f(m) = 0} satisfies the Universal Property of Kernels.
d) N/f(M) satisfies the Universal Property of Cokernels.
Let S be a set of n numbers. Let X be
the set of all subsets of S of size k, and let
Y be the set of all ordered k-tuples
(s1, s2, ,
sk)
such that
s1 < s2
< < sk.
That is,
X
=
{{s1, s2, ,
sk} | si S and all si's
are distinct}, and
Y
=
{(s1, s2, ,
sk) | si S and s1 <
s2 < < sk}.
(a) Define a one-to-one correspondence
f : X → Y.
Explain...
Let
R be the ring of all 2 x 2 real matrices.
A. Assume that A is an element of R such that AB=BA for all B
elements of R. Prove that A is a scalar multiple of the identity
matrix.
B. Prove that {0} and R are the only two ideals.
Hint: Use the Matrices E11, E12, E21, E22.
Let X be the set of all subsets of R whose complement is a
finite set in R:
X = {O ⊂ R | R − O is finite} ∪ {∅}
a) Show that T is a topological structure no R.
b) Prove that (R, X) is connected.
c) Prove that (R, X) is compact.