A ring R is subdirectly irreducible if the intersection of all nonzero ideals of R is nonzero.

Question:

(a) R is subdirectly irreducible if and only if whenever R is isomorphic to a subdirect product of {R_i Ii ϵ I}, R ≅ R_i for some i ε I.

(b) Every ring is isomorphic to a subdirect product of a family of subdirectly irreducible rings.

Fantastic news! We've Found the answer you've been seeking!

Step by Step Answer:

Related Book For book-img-for-question

Algebra Graduate Texts In Mathematics 73

ISBN: 9780387905181

8th Edition

Authors: Thomas W. Hungerford

See More Books

Question Posted: Aug 18, 2023 05:55 AM

See More Questions

An element a of a ring R is regular (in the sense of Von Neumann) if there exists x R such that axa = a. If every element of R is regular, then R is said to be a regular ring. (a) Every division...
Let R be any ring. The ascending chain condition (ACC) for ideals holds in R if every strictly increasing sequence N 1 N 2 N 3 of ideals in R is of finite length. The maximum condition (MC) for...
The radical of an ideal I in a ring R with identity is the intersection of all its minimal prime ideals [see Exercise 6]. Data from exercise 6 Let R have an identity. A prime ideal Pin R is called a...
How would this photo be different if the two people were both Americans?
The demand for subassembly S is 100 units in week 7. Each unit of S requires 1 unit of T and 2 units of U. Each unit of T requires 1 Unit of V, 2 units of W, and 1 unit of X. Finally, each unit of U...
What do the words completely randomized mean in an ANOVA?
The refrigerating capacity of 165 domestic refrigerators is approximately equal to: (a) 0.05 ton (b) 0.1 ton (c) 2 ton (d) 5 ton
Harry Plimpton is an accountant for Handley Company. Early this year, Harry made a highly favorable projection of sales and profits over the next 3 years for Handleys hot-selling computer PLEX. As a...
R is a division ring if and only if every element of R except one is left quasi-regular. Data from exercise 1 For each a, b R let a b = a+ b + ab. (a) o is an associative binary operation with...
Given that y = (1 + ln 4x) 3 /2 find the value of dy/dx at x = 1/4 e 3 .
Horned Frogs Fine Cooking maintains its books using cash-basis accounting. However, the company recently borrowed $50,000 from a local bank, and the bank requires the company to provide annual...
In 2014, a space probe approached the rocky core of the comet ChuryumovGerasimenko, which is only a few km in diameter. The probe then entered orbit around the comet at a distance of 30 km. The comet...
Marss moon Phobos orbits the planet at a distance of 9380 km from its center, and it takes 7 hours and 39 minutes to complete one orbit. What is the ratio of Marss mass to the mass of the earth?
We can model a small merry-go-round as a uniform circular disk with mass 88 kg and diameter 1.8 m. How many 22 kg children need to ride the merry-go-round, standing right at the outer edge, to double...
The engine in a small airplane is specified to have a torque of 500 N m. This engine drives a 2.0-m-long, 40 kg single-blade propeller. On startup, how long does it take the propeller to reach 2000...
A trap-jaw ant has mandibles that can snap shut with some force, as you might expect from its name. The formidable snap is good for more than capturing prey. When an ant snaps its jaws against the...
Two crystalline forms of white phosphorus are known. Both forms contain P4 molecules, but the molecules are packed together in different ways. The a form is always obtained when the liquid freezes....
On October 31 Juanita Ortega, owner of Outback Guide Service, received a bank statement dated October 30. Juanita found the following: 1. The checkbook has a balance of $2,551.34. 2. The bank...
Calculate the mean and standard deviation of household assets (ASSET). Assuming that this variable is positively skewed interpret the two statistics.
Find the mean and standard deviation of the incomes (INCOME) of the heads of households. We know that the distribution of income is extremely positively skewed. Briefly describe what the two...
Compute the mean and standard deviation of the ages (AGE) of the heads of households. Assuming that the distribution is bell shaped what do the mean and standard deviation tell you?
Ethan is playing baseball and is sliding into second base. He has a mass of 8 4 . 1 kg and begins sliding at a speed of 6 . 1 m / s . The average friction force acting against him during his motion...
300 FIS) 101b Ms=0.15 MK=0.1 Find work done by friction (ft-lb) after the block is pushed 6ft 6ft F 501b1 * ellipse 6ft S
What is the "right hand rule" for a fixed magnetic field and a moving charge (draw a picture)? Describe two situations where it can be applied

A ring R is subdirectly irreducible if the intersection of all nonzero ideals of R is nonzero.

Question:

Step by Step Answer:

a A ring R is subdirectly irreducible if and only if whenever R is isomorphic to a subdirect product ...View the full answer

Algebra Graduate Texts In Mathematics 73

Students also viewed these Mathematics questions