Generalized Hopfian modules

Strongly clean triangular matrix rings over local rings

“…Since x / ∈ Bl(J k ) ∪ Bl(1 − J k ) and J k ⊆ Bl(1 − J k ) (by B 2k and part (3) of Proposition 20) and therefore also (1 − J k ) ⊆ Bl(J k ), (6) implies that x ∈ Bl(J k−1 ) ∩ Bl(1 − J k−1 ). Since x / ∈ Bl(a k ) and x / ∈ Bl(b k ), we have a k / ∈ J k−1 and b k / ∈ (1 − J k−1 ).…”

“…In [6], Ghorbani and Haghany define a module M to be gH (generalized Hopfian) if the kernel of any surjective endomorphism of M is a small submodule of M. This property is shown to be equivalent to f −1 (X) being a small submodule of M for any small submodule X of M and any surjective endomorphism f of M. In a similar vein, one of the present authors (T. Dorsey) defines a ring R to be a gh-ring if, for every a, b ∈ R such that l a − r b is surjective on R Z(R) and for every small submodule X Z(R) ⊂ s R Z(R) , the preimage of X under l a − r b is small in R Z(R) . In [5], Dorsey shows that Theorem 33 remains true when h-ring is replaced by gh-ring.…”

Notes on Generalized Hopfian and Weakly Co-Hopfian Modules

“…Now one checks easily that R a R R since U X U U . 1 ⇒ 2 If A = a ij ∈ U = U R is a left unit, then a ii ∈ R i = 1 2 are left units in R, and R a ii R R. Obviously, [2] called such a ring R satisfying the above strengthening property a left domain and proved that R is a left domain if and only if for any 0 = x ∈ R and y ∈ R, then yx = x implies y is left invertible. The right domain is defined similarly.…”

“…For example, Hiremath [7], Varadarajan [10,11], Xue [13], Haghany [5], Liu [8], and Yang and Liu [14]. Recently Haghany and Vedadi [6], and Ghorbani and Haghany [2], respectively, introduced and investigated the weakly co-Hopfian and generalized Hopfian modules. The aim of this article is to study the generalized Hopficity and weakly co-Hopficity of modules over truncated polynomial and triangular matrix rings.…”

Polynomial rings over commutative reduced Hopfian local rings

“…The notion of Hopfian and co-Hopfian have been studied in the categories of groups, rings, modules and topological spaces. Since then the question has been generalized not only to other categories but also has been weakened and strengthened in quest of finding a classification by many researchers ( [2], [7], [8], [10], [11], [14]).…”