Question

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.

Answer #1

A
finitely generated module is projective if and only if it is a
direct summand of a finitely generated free module.

A
direct summand of a finitely generated module is finitely
generated.

Prove that every homomorphic image of a finitely generated
module is finitely generated.

Let R be a commutative ring with unity. Prove that the principal
ideal generated by x in the polynomial ring R[x] is a prime ideal
iff R is an integral domain.

View Z as a module over the ring R=Z[x,y] where x and
y act by 0. fond a free resolution of Z over R.

Let R be a commutative ring and let a ε R be a non-zero element.
Show that Ia ={x ε R such that ax=0} is an ideal of R. Show that if
R is a domain then Ia is a prime ideal

. Let M be an R-module; if me M let 1(m) = {x € R | xm = 0}.
Show that 1(m) is a left-ideal of R. It is called the order of m.
17. If 2 is a left-ideal of R and if M is an R-module, show that
for me M, λm {xm | * € 1} is a submodule of M.

Let G be a finitely generated group, and let H be normal
subgroup of G. Prove that G/H is finitely generated

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.

Let R be a commutative ring with unity. Let A consist of all
elements in A[x] whose constant term is equal to 0. Show that A is
a prime ideal of A[x]

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