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1
Integral Closure: Rees Algebras, Multiplicities, Algorithms
Show that m is of linear type if and only if R is a regular local ring. Exercise 1.174.
Let R be a Noetherian ring and I an ideal generated by a d-sequence x1 ,...,x n. If
grade I = r, show that x1 ,...,x r is a regular sequence. Exercise 1.175. Let R be ...
2
Algebraic Geometry: Part I: Schemes. With Examples and Exercises
A ring A satisfying these conditions is called a Jacobson ring. Exercise 10.16. A
scheme X is called a Jacobson scheme, if the subset of closed points is very
dense in X. (a) Show that for a scheme X the following are equivalent: (i) X is a ...
Ulrich Görtz, Torsten Wedhorn, 2010
3
Modules Over Discrete Valuation Domains
The function z — zq is a prime element of the ring. Exercise 6. Let R be a
commutative domain and let F be the field of fractions of R. The ring R is called
the valuation ring (of the field F) if either a G R or a-1 G R for every nonzero
element a in ...
Piotr A. Krylov, Askar A. Tuganbaev, 2008
4
Lattice-ordered Rings and Modules
... quotient ring of R. We will leave the verification of this to Exercise 13. If R is
reduced, then so is Q2(R) (Exercise 14) but Q(R) need not be reduced. In fact,
when R is a domain Q(R) is reduced precisely when it is a division ring (Exercise
26).
Stuart A. Steinberg, 2009
5
Commutative Algebra: Chapters 1-7
Let A be a principal ideal ring (Exercise 24), a a two-sided ideal ^0 of A and a an
element of a with the properties stated in Exercise 24 (f). Let P\M\ denote the
following property: M is a left A-module and every finitely generated submodule
of M ...
6
Topological Rings Satisfying Compactness Conditions
Any compact ring does not contain an infinite simple sub- ring. Exercise 6.2. If R
is a compact ring without nilpotents, then the quotient ring R/J(R) is a topological
product of finite fields. EXERCISE 6.3. Prove that if R is a finite ring with identity, ...
7
An Introduction to Noncommutative Noetherian Rings
The example fc[[x]] of a power series ring over a field shows that not all
semiprime (or even prime) rings are semiprimitive. Exercise 3Q. Find the
Jacobson radical of a formal triangular matrix ring Exercise 3R. Show that J(Mn(R
)) = Mn(J(R)) for ...
K. R. Goodearl, R. B. Warfield, Jr, 2004
8
Exercises in Classical
Ring Theory
Since R is a domain and hence a prime ring, Exercise 19 shows that R is left
primitive. Comment. The conditions (1), (2), (3) above are satisfied by. for some j.
Now Mj/Mj−1 is a simple faithful left R-module, so R is left primitive. Solution.
9
A Logical Introduction to Proof
Construct and evaluate the quotient ring Exercise Notes: For Exercise 5, review
Lemma 8.3.13. For Exercise 10, look over Exercise 13 on page 260 and Theorem
8.6.9. For Exercise 16, review the proof of Theorem 8.6.9. Core Concepts in ...
Daniel W. Cunningham, 2012
10
Interactions Between Homotopy Theory and Algebra: Summer ...
... injective hull of the residue field of R (as graded ring). EXERCISE 18. Let x be
a non-zerodivisor in a local Noetherian ring R. Directly from Definition 4.1, prove
that R is Cohen-Macaulay if and only if R/Rx is Cohen- Macaulay. EXERCISE 19.
Luchezar L. Avramov, 2007
