Additive Representations of Elements in Rings: A Survey

Srivastava, Ashish K.

doi:10.1007/978-981-10-1651-6_4

Cited by 3 publications

(2 citation statements)

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“…There are several equivalent ways to define the property of an algebra to be generated by units: as an algebra, as a ring, as a vector space (that is, A = span(U(A)), and as an additive group (which means that every element of A is a finite sum of units). For two surveys on rings generated by units, see [18] and [19,Section 1].…”

Section: The Gk ż Property Of Algebras and Generation By Unitsmentioning

confidence: 99%

On the Gleason-Kahane-Żelazko Theorem for Associative Algebras

Roitman

Sasane

2022

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The classical Gleason-Kahane-Żelazko Theorem states that a linear functional on a complex Banach algebra not vanishing on units, and such that $$\Lambda (\textbf{1})=1$$ Λ ( 1 ) = 1 , is multiplicative, that is, $$\Lambda (ab)=\Lambda (a)\Lambda (b)$$ Λ ( a b ) = Λ ( a ) Λ ( b ) for all $$a,b\in A$$ a , b ∈ A . We study the GKŻ property for associative unital algebras, especially for function algebras. In a GKŻ algebra A over a field of at least 3 elements, and having an ideal of codimension 1, every element is a finite sum of units. A real or complex algebra with just countably many maximal left (right) ideals, is a GKŻ algebra. If A is a commutative algebra, then the localization $$A_{P}$$ A P is a GKŻ-algebra for every prime ideal P of A. Hence the GKŻ property is not a local-global property. The class of GKŻ algebras is closed under homomorphic images. If a function algebra $$A\subseteq {\mathbb {F}}^{X}$$ A ⊆ F X over a subfield $${\mathbb {F}}$$ F of $${\mathbb {C}}$$ C , contains all the bounded functions in $${\mathbb {F}}^{X}$$ F X , then each element of A is a sum of two units. If A contains also a discrete function, then A is a GKŻ algebra. We prove that the algebra of periodic distributions, and the unitisation of the algebra of distributions with support in $$(0,\infty )$$ ( 0 , ∞ ) satisfy the GKŻ property, while the algebra of compactly supported distributions does not.

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Section: The Gk ż Property Of Algebras and Generation By Unitsmentioning

confidence: 99%

On the Gleason-Kahane-Żelazko Theorem for Associative Algebras

Roitman

Sasane

2022

Results Math

View full text Add to dashboard Cite

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“…For example, clean rings are rings where every element is uniquely the sum of an idempotent and a unit. The article [9] gives a survey on recent results on additive representations of ring elements. In this regard, it will be useful to have a good knowledge of rings with idempotents.…”

Section: Introductionmentioning

confidence: 99%