Kasch ring

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 :

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

Examples

The content below can be found in references such as , , .

• 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: :: \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. \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

• {{citation |last=Kasch |first=Friedrich |title=Grundlagen einer Theorie der Frobeniuserweiterungen

References

1. This ideal is necessarily a [[minimal right ideal]].
2. {{harv. Kasch. 1954
3. {{harv. Morita. 1966