Find all the units of
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Let R be a commutative ring with unity and r ∈ R . Prove that using the Fundamental Theorem of Homomorphism. Hence show that
Let
i) Show that M is an ideal of R .
ii) Show that if a|5 / or b/|5 , then 5| for ,a . b ∈
iii) Hence show that if N is an ideal of R properly containing M , then N = R .
iv) Show that is a field, and give two distinct non-zero elements of this field.
Let be a finite abelian group and
Prove that
Let Check whether D is a UFD or not.
Let G be a group of order n ≥ 2 , with only two subgroups and itself. Find a minimal generating set for G . Also, find out whether n is a prime or a composite number, or can be either.
Explicitly give the elements and structure of the group
Which of the following statements are true? Justify your answers. (This means that if you think a statement is false, give a short proof or an example that shows it is false. If it is true, give a short proof for saying so.)
i) where
is the Euler-phi function.
ii) If and
are groups, and
is a group homomorphism, then
iii) If G is an abelian group, then G is cyclic.
iv) If G is a group and ,then
v) Every element of has order at most n .
vi) If R is a ring and I is an ideal of R , then I xr = rx ∀ x ∈ I and r ∈ R .
vii) If is a product of an even number of disjoint cycles, then sign
viii) If a ring has a unit, then it has only one unit.
ix) The characteristic of a finite field is zero.
x) The set of discontinuous functions from to
form a ring with respect to pointwise addition and multiplication.