In this paper, we extend the recently introduced vertex-degree-based topological index, the Sombor index, and we call it general Sombor index. The general Sombor index generalizes both the forgotten index and the Sombor index. We present the bounds in terms of other important graph parameters for general Sombor index. We also explore the Nordhaus–Gaddum-type result for the general Sombor index. We present further the relations between general Sombor index and other generalized indices: general Randić index and general sum-connectivity index.

A ring R is called a left (right) SF-ring if simple left (right) iî-modules axe flat. It is still unknown whether a left (right) SF-ring is von Neumann regular. In this paper, we give some conditions for a left (right) SF-ring to be (a) von Neumann regular; (b) strongly regular; (c) division ring. It is proved that: (1) a right SF-ring R is regular if maximal essential right (left) ideals of R are weakly left (right) ideals of R (this result gives an affirmative answer to the question raised by Zhang in 1994); (2) a left SF-ring R is strongly regular if every non-zero left (right) ideal of R contains a non-zero left (right) ideal of R which is a W-ideal; (3) if iî is a left SF-ring such that Z(x) (r(a;)) is an essential left (right) ideal for every right (left) zero divisor x of R, then ñ is a division ring.

In this paper we show that for any arbitrary graph E and for a field K, principally-injective conditions for the Leavitt path algebra L K (E) are equivalent to that the graph E being acyclic. We also show that the principally-injective Leavitt path algebras are precisely the von Neumann regular Leavitt path algebras.

In this paper we show that for any arbitrary graph E and for a field K, principally-injective conditions for the Leavitt path algebra L K (E) are equivalent to that graph E being acyclic. We also show that the principally-injective Leavitt path algebras are precisely the von Neumann regular Leavitt path algebras.

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