Question

Let
m,n be integers. show that the intersection of the ring generated
by n and the ring generated by m is the ring generated by their
least common multiple.

Answer #1

Let
m and n be positive integers and let k be the least common multiple
of m and n. Show that mZ intersect nZ is equal to kZ. provide
justifications pleasw, thank you.

Let
m and n be positive integers and let k be the least common multiple
of m and n. Show that mZ intersect nZ is equal to kZ. provide
justifications please, thank you.

Let m,n be any positive integers. Show that if m,n have no
common prime divisor (i.e. a divisor that is at the same time a
prime number), then m+n and m have no common prime divisor. (Hint:
try it indirectly)

Let R be a ring.
Show that R[x] is a finitely generated R[x]-module if and only if
R={0}.
Show that Q is not a finitely generated Z-module.

Let
m and n be positive integers. Exhibit an arrangement of the
integers between 1 and mn which has no increasing subsequence of
length m + 1, and no decreasing subsequence of length n + 1.

Let R be a ring and let M and N be
right R-modules. Assume that the only
R-homorphisms M → N and N →
M are 0 maps. Prove that
EndR(M⊕N) ∼=
EndR(M) ⊕ EndR(N) (direct
sum of rings). Remember the convention used for composition of
R-homomorphismps.

Let a and b be non-zero integers. Do not appeal to the
fundamental theorem of arithmetic to do to this problem.
Show that if a and b have a least common multiple it is
unique.

Let I, M be ideals of the commutative ring R. Show that M is a
maximal ideal of R if and only if M/I is a maximal ideal of
R/I.

1. A) Show that the set of all m by n matrices of integers is
countable where m,n ≥ 1 are some ﬁxed positive integers.

Write Java program Lab42.java which takes as input two positive
integers m and n and computes their least common multiple
by calling method lcm(m,n), which in turn calls recursive method
gcd(m,n) computing the greatest common divisor of m and n.
Recall, that lcm(m,n) can be defined as quotient of m * n and
gcd(m,n).

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