WE INVESTIGATE THE CLASS OF PIECEWISE PRIME, PWP, A RING WHICH PROPERLY INCLUDES ALL PIECEWISE DOMAIN. FOR A PWP RING WE DETERMINE A LARGE CLASS OF RING EXTENSIONS WHICH HAVE A GENERALIZED triangular matrix REPRESENTATION FOR WHICH THE DIAGONAL RINGS ARE PRIME. WE INVESTIGATE THE QUASI-BAER AND RELATED CONDITIONS ON MONOID RINGS R[G]. IN PARTICULAR WE CONSIDER THE TRANSFER OF THE QUASI-BAER AND RELATED CONDITIONS BETWEEN R AND A U.P.-MONOID RING R[G] OF A U.P.-MONOID G OVER R. WE CHARACTERIZE THE SEMI-CENTRAL IDEMPOTENTS OF VARIOUS RING EXTENSIONS OF R IN TERMS OF THE SEMI-CENTRAL IDEMPOTENTS OF R.

LET R BE A ASSOCIATIVE RING WITH IDENTITY. WE PROVE THAT FOR (FORMULA), IF A0 OR 1- A0 IS STRONGLY P-REGULAR IN R, THEN (A0, A1, · · ·, AN−1) IS A STRONGLY CLEAN ELEMENT IN THE triangular matrix RING (FORMULA). AS A COROLLARY, WE DEDUCE THAT IF R IS A STRONGLY P-REGULAR RING, THEN (FORMULA) IS A STRONGLY CLEAN RING. WE ALSO SHOW THAT THE (K, G (X)) -CLEAN PROPERTY OF A RING R AND (FORMULA) IS EQUIVALENT.

A REAL matrix R WITH NONNEGATIVE ENTRIES IS CALLED ROW SUB STOCHASTIC IF EACH ROW SUM ISAT MOST 1. FOR X, Y Î RN, WE SAY THAT X IS ROW SUB STOCHASTIC upper triangular MAJORIZED BY Y (WRITE AS X≺SUT Y) IF THERE EXISTS A ROW SUB STOCHASTIC upper triangular matrix R SUCH THAT X=RY. IN THIS PAPER, THE STRUCTURE OF ALL LINEAR FUNCTIONS T: R3 → R3, PRESERVING (STRONGLY PRESERVING)≺SUT ARE CHARACTERIZED.

IN THIS PAPER, WE STATE AND PROVE A NEW FORMULA EXPRESSING EXPLICITLY THE RIEMANN-LIOUVILLE FRACTIONAL INTEGRAL OF THE triangular FUNCTIONS (TFS) WITH ANY FRACTIONAL-ORDER, IN TERMS OF TFS THEMSELVES AND USE IT TO CONSTRUCT A NEW AND GENERAL FORMULATION FOR THE triangular FUNCTION (TF) OPERATIONAL matrix OF FRACTIONAL INTEGRAL (TF-OMFI). ALSO, BY USING THE TF-OMFI AND A SPECTRAL TAU METHOD, WE DEVELOP A DIRECT SOLUTION TECHNIQUE FOR SOLVING A NONLINEAR FRACTIONAL DIFFERENTIAL EQUATION.