We investigate partial cancellation of modules and show that if an ideal I of an exchange ring R has stable range one, then A⊕B≌A⊕C implies B≌C for all A∈FP (I). The converse is true when R is a regular ring. For an ideal I of a regular ring, we also show that I has stable range one if and only if perspectivity is transitive in L(A) for all A∈ FP (I). These give nontrivial generalizations for unit-regularity.
] Slowikowski , W . and Zawadowski , W . A generalization of maximal ideals method of stone and Gel’fand . Fund. Math . 42 ( 1995 ), 225 – 231 .  Vandiver , H. S . Note on a simple type of algebra in which the cancellation law of addition does not
Let R be a ring and define x ○ y = x + y - xy, which yields a monoid (R, ○), called the circle semigroup of R. This paper investigates the relationship between the ring and its circle semigroup. Of particular interest are the cases
where the semigroup is simple, 0-simple, cancellative, 0-cancellative, regular, inverse, or the union of groups, or where
the ring is simple, regular, or a domain. The idempotents in R coincide with the idempotents in (R, ○) and play an important role in the theory developed.
Let R be a domain with quotient field K. It is proved that R is an integrally closed domain if and only if every nonzero t-ideal of R is complete, if and only if every nonzero v-ideal of R is complete. We also obtain that every prime ideal of an integrally closed domain is integrally closed, and every strongly prime ideal of a domain is integrally closed. Moreover, we introduce the notion of w-cancellation ideals and give some equivalent characterizations of PVMDs. In particular, it is proved that R is a PVMD if and only if every w-ideal of R is complete.
1,10-Phenanthroline (Phen) as a new additive was added into the solutions of KH2PO4 (KDP) and NH4H2PO4 (ADP) in a small amount (∼2.5·10−3 M L−1). The crystals were grown from the aqueous solutions of pH ∼4.5 at constant temperature by solvent evaporation technique.
It leads to an increase in metastable zone width and assists the bulk growth process. The growth rate of crystals in the presence
of Phen decreases considerably with an increase in impurity concentration (∼2.5·10−2 M L−1). Not much variation is observed in FTIR and XRD of pure and doped ADP/KDP. It appears that the growth promoting effect (GPE)
of Phen is caused by the adsorption of the organic additive on the prism faces of ADP/KDP crystals. Higher optical transmittance
is observed in the presence of the dopant. Detailed microhardness studies of ADP crystals reveal the anisotropy in the hardness
behaviour. Scanning electron microscope (SEM) photographs exhibit the effectiveness of the impurity in changing the surface
morphology of ADP/KDP crystals. Contrary to expectations, Phen depresses the NLO efficiency of ADP/KDP, suggesting that the
molecular alignments in the presence of Phen results in cancellation effects disturbing the non-linearity.