Part Algebra of Qualifying Examination, August 23, 2010
Instructions: Do all questions, justify your answers with the necessary proofs. All
rings are associative (not necessarily commutative) with identity, and all modules are left
unitary modules. We denote by Z the ring of integers, and by R, Q the elds of real and
rational numbers, respectively.
1. Given a ring R, the opposite ring Rop has the same underlying abelian group as R
and a new multiplication dened by a b = ba, for all a, b R, where juxstaposition
denotes the original multiplication in R. Let Mm,n (R) be the set of m n matrices with
entries from R. The set Mn (R) = Mn,n (R) is a ring with respect to the usual addition
and multiplication of matrices. The sets Mn (R) and Mn (Rop ) coincide as abelian groups,
but have dierent ring structures. For all A Mm,n (R), a R, we write a A = Aa
and A a = aA for the scalar multiplication. For all B Mn,p (R), we write AB for the
product of A and B over R, and A B for their product over Rop .
(a) Consider R as an R-module and prove the following.
(i) (1 point) For any a R, the map a : R R given by a (r) = a r, for all r R,
is an endomorphism of R (as an R-module).
(ii) (2 points) For any EndR (R), there is a unique a R satisfying (r) = a r,
for all r R.
(iii) (2 points) The map Rop EndR (R) given by a a is an isomorphism of rings.
(b) Consider L = Mn,1 (R) as an R-module and prove the following.
(i) (1 point) For any A Mn (R), the map A : L L given by A (X ) = A X, for
all X L, is an endomorphism of L.
(ii) (2 points) For any EndR (L), there is a unique A Mn (R) satisfying (X ) =
A X, for all X L.
(iii) (2 points) The map Mn (Rop ) EndR (L) given by A A is an isomorphism of
rings.
(c) Consider L as an Mn (R)-module and prove the following.
(i) (2 points) For any a R, the map a : L L given by a (X ) = a X, for all
X L, is an endomorphism of L.
(ii) (4 points) For any EndMn (R) (L), there is a unique a R satisfying (X ) =
a X, for all X L.
(iii) (2 points) The map Rop EndMn (R) (L) given by a a is an isomorphism of
rings.
(d) points) (2 Consider N = Mn,p (R) as an Mn (R)-module. State, but DO NOT prove
the analogs of (i)(iii) of part (c).
2. Let R be an arbitrary ring.
(a) (2 points) Give the denition of when an R-module P is projective.
(b) (2 points) Do projective R-modules exist? You may quote an appropriate theorem.
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Algebra Part of Qualifying Examination, August 23, 2010, Page 2
2. (continued)
(c) (4 points) If P is a projective R-module, prove that an exact sequence of R-modules
of the form 0 X Y P 0 splits. Begin with a denition of a split exact sequence.
(d) (5 points) Prove that an R-module A is projective if and only if for each exact
sequence 0 X Y Z 0 of R-modules, the sequence
0 HomR (A, X )
HomR (A,)
HomR (A, Y )
HomR (A, )
HomR (A, Z ) 0
of abelian groups is exact. You may use the denition of a projective module and the left
exactness of the functor Hom.
3. Let R be an arbitrary ring.
(a) (2 points) Give the denition of when an R-module I is injective.
(b) (2 points) State Baers criterion for when an R-module I is injective.
(c) (6 points) Using Baers criterion, prove that if R is a (commutative) principal ideal
domain (PID), then an R-module A is injective if and only if it is divisible. First give a
denition of a divisible module.
(d) (2 points) Do divisible modules over a PID exist? Explain.
(e) (2 points) Explain how one can construct an injective module over an arbitrary ring
R using a divisible Z-module. Quote an appropriate statement.
R0
of all 2 2 matrices A = (aij )1i,j 2 satisfying
RQ
Q, and a12 = 0, with the usual operations of matrix addition and
4. Consider the ring R =
a11 , a21 R, a22
multiplication.
(a) (1 points) Is the ring R left artinian?
(b) (1 points) Is the ring R left noetherian?
(c) (1 points) Is the ring R right artinian?
(d) (1 points) Is the ring R right noetherian?
(e) (5 points) Find the radical of R, J (R), and describe the ring structure of R/J (R)
in terms of R and Q.
(f) (4 points) Describe the nonisomorphic simple left R-modules by indicating their
underlying abelian group and R-action.
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