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...

Given an n×n matrix c over a unitary ring R, the centralizer of in full Mn(R) is called principal ring, denoted by Sn(c,R). We investigate its structure and prove: (1) If invertible with c-free point, or if R has no zero-divisors Jordan-similar all eigenvalues center then separable Frobenius extension Sn(c,R) sense Kasch. (2) integral domain matrix, cellular R-algebra Graham Lehrer. In particul...

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. ...

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 ...

The notion of FTF ring (see Definition 1.1 or Proposition 1.2) captures homological and finiteness properties shared by several classes of rings. Thus coherent rings with left flat-dominant dimension ≥ 1 [3, Corolario 2.2.11] or rings having quasi-Frobenius two-sided maximal quotient ring [7, Proposition 3.6; 3, Teorema 2.3.10] are examples of FTF rings. Moreover, FTF ring and QF-3 ring are rel...