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STRONGLY CLEAN MATRIX RINGS OVER COMMUTATIVE RINGS
, 2008
"... A ring R is called strongly clean if every element of R is the sum of a unit and an idempotent that commute. By SRC factorization, Borooah, Diesl, and Dorsey [3] completely determined when Mn(R) over a commutative local ring R is strongly clean. We generalize the notion of SRC factorization to com ..."
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A ring R is called strongly clean if every element of R is the sum of a unit and an idempotent that commute. By SRC factorization, Borooah, Diesl, and Dorsey [3] completely determined when Mn(R) over a commutative local ring R is strongly clean. We generalize the notion of SRC factorization to commutative rings, prove that commutative nSRC rings (n ≥ 2) are precisely the commutative local rings over which Mn(R) is strongly clean, and characterize strong cleanness of matrices over commutative projectivefree rings having ULP. The strongly πregular property (hence, strongly clean property) of Mn(C(X, C)) with X a Pspace relative to C is also obtained where C(X, C) is the ring of complex valued continuous functions.
Various Notions of Compactness
, 2012
"... We document various notions of compactness, with some of their useful properties. Our main reference is [Engelking, 1989]. For counterexamples, refer to [Steen and Seebach Jr., 1995] while for background, see the excellent textbooks [Willard, 2004] or [Munkres, 2000]. Some of the proofs are taken fr ..."
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We document various notions of compactness, with some of their useful properties. Our main reference is [Engelking, 1989]. For counterexamples, refer to [Steen and Seebach Jr., 1995] while for background, see the excellent textbooks [Willard, 2004] or [Munkres, 2000]. Some of the proofs are taken freely from the internet. The writer expresses his gratitude to all sources. 1
The Differentiability of the Upper Envelop
, 2012
"... We present the proof of the DanskinValadier theorem, i.e. when the directional derivative of the supremum of a collection of functions admits a natural representation. 1 Preliminary Consider a collection of extended realvalued functions fi: X 7 → R̄, where i ∈ I is some index set, X is some real v ..."
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We present the proof of the DanskinValadier theorem, i.e. when the directional derivative of the supremum of a collection of functions admits a natural representation. 1 Preliminary Consider a collection of extended realvalued functions fi: X 7 → R̄, where i ∈ I is some index set, X is some real vector space, and R ̄: = R ∪ {±∞}. Define the supremum (i.e. upper envelop) of the collection as f(x): = sup i∈I fi(x). (1) We are interested in studying the directional derivative of f, hopefully relating it to the directional derivatives of fi. Recall that the directional derivative of g, along direction d, is defined as g′(x; d): = lim t↓0 g(x+ td) − g(x)