If X is a Tychonoff space, C(X) its ring of real-valued continuous functions. In this paper, we study non-essential ideals in C(X). Let a be a infinite cardinal, then X is called a-Kasch (resp. ā-Kasch) space if given any ideal (resp. z-ideal) I with gen (I) < a then I is a non-essential ideal. We show that X is an א0-Kasch space if and only if X is an almost P -space and X is an א1-Kasch space...

Let $R$ be a ring and $M$ a right $R$-module. We call $M$, coretractable relative to $overline{Z}(M)$ (for short, $overline{Z}(M)$-coretractable) provided that, for every proper submodule $N$ of $M$ containing $overline{Z}(M)$, there is a nonzero homomorphism $f:dfrac{M}{N}rightarrow M$. We investigate some conditions under which the two concepts coretractable and $overline{Z}(M)$-coretractable...

A right Johns ring is a Noetherian in which every ideal annihilator. It known that RR the Jacobson radical J(R)J(R) of nilpotent and Soc(R)(R) an essential RR. Moreover, Kasch, is, simple RR-module can be embedded For M∈RM∈R-Mod we use concept MM-annihilator define module (resp. quasi-Johns) as MM such submodule MM-annihilator. called quasi-Johns if any set submodules satisfies ascending chain ...

Seung-kook YUN** , Seong-Sik YOON** , Sungchul KANG**, and Munsang KIM** **Intelligent Robotics Research Center, Korea Institute of Science and Technology, Hawolgok-dong1 39-1, Sungbuk-ku, Seoul 136-791, Korea E-mail: yunsk@mit.edu, yoon@cello.korea.ac.kr, {kasch, munsang}@kist.re.kr Mr. Seung-kook Yun is now with Computer Science and Artificial Intelligence Laboratory, Massachusetts Institute ...

5. L. K. Hua, A note on the total matrix ring over a non-commutative field, Annales de la Société Polonaise de Mathématique vol. 25 (1952) pp. 188-198. 6. N. Jacobson and C. E. Rickart, Jordan homomorphisms of rings, Trans. Amer. Math. Soc. vol. 69 (1950) pp. 479-502. 7. F. Kasch, Invariante Untermoduln des Endomorphismenrings eines Vektorraums, Archiv der Mathematik vol. 4 (1953) pp. 182-190. ...