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Let R R be a ring and A ( R ) = { x ∈ R : x A(R) = \{ x \in R:x belongs to the second commutant of { x n , x n + 1 } \{ {x^n},{x^{n + 1}}\} for all integers n > 1 } n > 1\} . It is shown that in a prime ring R , A ( R ) = R R,A(R) = R if and only if R R has no nilpotent elements. The set A ( U ) A(U) is studied for some special ∗ \ast -algebras. It is shown that the normal elements of a proper ∗ \ast -algebra U U belong to A ( U ) A(U) . If U U is also prime then A ( U ) = { x ∈ U : x A(U) = \{ x \in U:x belongs to the second commutant of { x n , x n + 1 } \{ {x^n},{x^{n + 1}}\} for some n > 1 } n > 1\} . The set A ( B ( H ) ) A(B(H)) is studied, where B ( H ) B(H) is the algebra of bounded operators on a Hilbert space H H . Necessary and sufficient conditions for some special types of operators to belong to A ( B ( H ) ) A(B(H)) are obtained.

Sludge formed downhole in the production interval can be classified into crude oil-based or mud-based sludge. Sludge obstruction may result in partial or total loss of well productivity. Oil-based mud is commonly used in drilling of the pay zones in sandstone formations as a less/non damaging fluid. Oil-based mud typically contain emulsifier, viscosifer and other additives including polymer blend and calcium carbonate to serve different functions. Presence of emulsifier may increase emulsion tendency upon interaction with downhole environment. The resulting emulsion might be tight to an extent that a thick sludge is formed which can impair well productivity. Similarly, oil-based sludge may form from oil/water interaction in presence of emulsifiers, asphaltene, wax, solids, shear, etc. Identification of the sludge material will help in development of an effective chemical treatment to remove formation damage and restore well productivity. In this study, an extensive laboratory work was conducted to explore potential interactions of different downhole environment contaminants/factors on formation of oil-based and mud-based sludge. Typical mud-based and crude oil-based sludge samples were characterized using different analytical techniques including solvent extraction, XRD, TGA, ICP and viscosity. The results showed that the mud-based sludge sample contained calcium carbonate, dolomite quartz as the main components in the inorganic phase while the organic phase include polymers and oil. The oil-based sludge sample contained mainly water (82 wt%) with some solid particulates and asphaltene in the organic phase. Analysis of supernatants generated from solubility tests conducted for the mud-based and oil-based sludge samples revealed in addition to the high amount of calcium presence of iron in considerable amount (nearly 1,000 mg/L). Interaction of ferric chloride, quartz with an invert-emulsion mud was investigated. A significant increase in viscosity was observed upon incorporation of these contaminates with the mud sample. Iron ions in the aqueous phase tended to stabilized emulsion. This paper presents in detail mechanisms of mud-based and crude oil-based sludge formation upon interaction with environment. It also examined several chemical formulations for removal of mud-based and oil-based sludge samples.

Some conditions which are equivalent to finite dimensionality for a semi-simple Banach algebra A are given, and simplification of the proofs of previously known results are included.These conditions include local finiteness, spectral finiteness and the condition that socle A exists and is equal to A.

ABSTRACT. Given a Banach algebra A, the compactum of A is defined to be the set of elements x g A such that the operator a xax is compact. General properties of the compactum and its relation to the socle of A are discussed. Characterizations of finite dimensionality of a seml-slmple Banach algebra are given in terms of the compactum and the socle of A.