Kasch ring

Key takeaways

Kasch ring belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Kasch ring to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Kasch ring from memory before moving on to harder problems.

Reference excerpt

In ring theory, a subfield of abstract algebra, a right Kasch ring is a ring R for which every simple right R-module is isomorphic to a right ideal of R. Analogously the notion of a left Kasch ring is defined, and the two properties are independent of each other. Kasch rings are named in honor of mathematician Friedrich Kasch. Kasch originally called Artinian rings whose proper ideals have nonzero annihilators S-rings. The characterizations below show that Kasch rings generalize S-rings.

Definition Equivalent definitions will be introduced only for the right-hand version, with the understanding that the left-hand analogues are also true. The Kasch conditions have a few equivalent statements using the concept of annihilators, and this article uses the same notation appearing in the annihilator article. In addition to the definition given in the introduction, the following properties are equivalent definitions for a ring R to be right Kasch. They appear in Lam (1999, p. 281):

For every simple right R-module M, there is a nonzero module homomorphism from M into R. The maximal right ideals of R are right annihilators of ring elements, that is, each one is of the form r . a n n ( x ) {\displaystyle \mathrm {r.ann} (x)} where x is in R. For any maximal right ideal T of R, ℓ . a n n ( T ) ≠ { 0 } {\displaystyle \mathrm {\ell .ann} (T)\neq \{0\}} . For any proper right ideal T of R, ℓ . a n n ( T ) ≠ { 0 } {\displaystyle \mathrm {\ell .ann} (T)\neq \{0\}} . For any maximal right ideal T of R, r . a n n ( ℓ . a n n ( T ) ) = T {\displaystyle \mathrm {r.ann} (\mathrm {\ell .ann} (T))=T} . R has no dense right ideals except R itself.

Examples The content below can be found in references such as Faith (1999, p. 109), Lam (1999, §§8C,19B), Nicholson & Yousif (2003, p.51).

Let R be a semiprimary ring with Jacobson radical J. If R is commutative, or if R/J is a simple ring, then R is right (and left) Kasch. In particular, commutative Artinian rings are right and left Kasch. For a division ring k, consider a certain subring R of the 4-by-4 matrix ring with entries from k. The subring R consists of matrices of the following form:

[ a 0 b c 0 a 0 d 0 0 a 0 0 0 0 e ] {\displaystyle {\begin{bmatrix}a&0&b&c\\0&a&0&d\\0&0&a&0\\0&0&0&e\end{bmatrix}}}

This is a right and left Artinian ring which is right Kasch, but not left Kasch. Let S be the ring of power series on two noncommuting variables X and Y with coefficients from a field F. Let the ideal A be the ideal generated by the two elements YX and Y 2. The quotient ring S/A is a local ring which is right Kasch but not left Kasch. Suppose R is a direct product of infinitely many nonzero rings labeled Ak. The direct sum of the Ak forms a proper ideal of R. It is easily checked that the left and right annihilators of this ideal are zero, and so R is not right or left Kasch. The 2-by-2 upper (or lower) triangular matrix ring is not right or left Kasch. A ring with right socle zero (i.e. s o c ( R R ) = { 0 } {\displaystyle \mathrm {soc} (R_{R})=\{0\}} ) cannot be right Kasch, since the ring contains no minimal right ideals. So, for example, domains which are not division rings are not right or left Kasch.

References

Faith, Carl (1999), Rings and things and a fine array of twentieth century associative algebra, Mathematical Surveys and Monographs, vol. 65, Providence, RI: American Mathematical Society, pp. xxxiv+422, ISBN 978-0-8218-0993-8, MR 1657671 Kasch, Friedrich (1954), "Grundlagen einer Theorie der Frobeniuserweiterungen", Math. Ann. (in German), 127: 453–474, doi:10.1007/bf01361137, ISSN 0025-5831, MR 0062724 Lam, Tsit-Yuen (1999), Lectures on modules and rings, Graduate Texts in Mathematics No. 189, Berlin, New York: Springer-Verlag, ISBN 978-0-387-98428-5, MR 1653294 Morita, Kiiti (1966), "On S-rings in the sense of F. Kasch", Nagoya Math. J., 27 (2): 687–695, doi:10.1017/S0027763000026477, ISSN 0027-7630, MR 0199230 Nicholson, W.K.; Yousif, M.F. (2003), Quasi-Frobenius rings, Cambridge Tracts in Mathematics, vol. 158, Cambridge: Cambridge University Press, pp. xviii+307, doi:10.1017/CBO9780511546525, ISBN 978-0-521-81593-2, MR 2003785

Worked examples

Example 1 — a first encounter with Kasch ring

Start with the simplest possible case. Write down what Kasch ring claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Kasch ring before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Kasch ring ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Kasch ring

In research: Kasch ring appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Kasch ring in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Kasch ring is common in secondary-school and first-year university syllabi. It links to neighbouring topics Algebraic structures, Ring theory, so understanding it makes those chapters shorter.
In everyday life: Look for Kasch ring outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study Kasch ring in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Kasch ring means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Kasch ring out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Kasch ring in simple terms?

In ring theory, a subfield of abstract algebra, a right Kasch ring is a ring R for which every simple right R-module is isomorphic to a right ideal of R. Analogously the notion of a left Kasch ring is defined, and the two properties are independent of each other.

Why does Kasch ring matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Kasch ring?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Kasch ring.