Let A be such that its only right ideals are {¯0} (neutral
element) and A. Show...
Let A be such that its only right ideals are {¯0} (neutral
element) and A. Show that A or is a ring with division or
A is a ring with a prime number of elements in which a · b = 0 for
any, b ∈ A.
Let
I1, I2 be ideals of R and J1, J2 be ideals of S. Show that (I1 +
I2)^extension= I1^extension + I2^extension where I1, I2 are
contained in R
|^e
is defined as the extension of I to S: Let R and S be commutatuve
ring and f:R to S be a ring homomorphism. For each ideal I of R,
the ideal f(I)S of S generated by f(I) is the extension of I to
S.
Let a be a positive element in an ordered field. Show that if n
is an odd number, a has at most one nth root; if n is an
even number, a has at most two nth roots.
Let us consider a random variable X is the element of U(0, a) so
that a has been obtained as a sample from a random variable A which
follows a uniform distribution A is the element of U(0, l) with
known parameter value l.
Estimates of a based on 1) the method of moments, 2) the method
of maximum likelihood and 3) the Bayesian-based methods,
respectively in R.
Read a data sample of r.v. X from the file
sample_x.csv. Estimate...
Let a and b be integers and consider (a) and (b) the ideals they
generate. Describe the intersection of (a) and (b), the product of
(a) and (b), the sum of (a) and (b) and the Ideal quotient
(aZ:bZ).
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).
Let B be a finite commutative group without an element of order
2. Show the mapping of b to b2 is an automorphism of B. However, if
|B| = infinity, does it still need to be an automorphism?
show that a 2x2 complex matrix A is nilpotent if and only if
Tr(A)=0 and Tr(A^2)=0. give an example of a complex 2x2 matrix
which is not nilpotent but whose trace is 0