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.
2 Let F be a field and let R = F[x, y] be the ring of
polynomials in two variables with coefficients in F.
(a) Prove that
ev(0,0) : F[x, y] → F
p(x, y) → p(0, 0)
is a surjective ring homomorphism.
(b) Prove that ker ev(0,0) is equal to the ideal (x, y) = {xr(x,
y) + ys(x, y) | r,s ∈ F[x, y]}
(c) Use the first isomorphism theorem to prove that (x, y) ⊆
F[x, y]...
Let f(t) =t^2−1 and g(t) =e^t.
(a) Graph f(g(t)) and g(f(t)).
(b) Which is larger,f(g(5)) or g(f(5))? Justify your answer.
(c) Which is larger, (f(g(5)))′or g(f(5))′? Justify your
answer.
a) Let S ⊂ R, assuming that f : S → R is a continuous function,
if the
image set {f(x); x ∈ S} is unbounded prove that S is unbounded.
b) Let f : [0, 100] → R be a continuous function such that f(0) =
f(2),
f(98) = f(100) and the function g(x) := f(x+ 1)−f(x) is equal to
zero in at most
two points of the interval [0, 100].
Prove that (f(50) − f(49))(f(25) − f(24)) >...
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.