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<p>Let <InlineEquation ID="IEq4"> <EquationSource Format="TEX">\(\mathcal {R}\)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="script">R</mi> </math></EquationSource> </InlineEquation> be a 2-torsion free unital <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(*\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow /> <mo>∗</mo> </mrow> </math></EquationSource> </InlineEquation>-ring with a non-trivial symmetric idempotent. In this study, we show that under some mild assumptions, if a map <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\( \Delta \)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Δ</mi> </math></EquationSource> </InlineEquation> : <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\(\mathcal {R}\rightarrow \mathcal {R}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">R</mi> <mo stretchy="false">→</mo> <mi mathvariant="script">R</mi> </mrow> </math></EquationSource> </InlineEquation> (not necessarily additive) satisfies <Equation ID="Equ17"> <EquationSource Format="TEX">\(\begin{aligned} \Delta ( U \circledast K \circ L) = \Delta (U ) \circledast K \circ L +U \circledast \Delta ( K) \circ L+ U \circledast K \circ \Delta ( L) \end{aligned}\)</EquationSource> <EquationSource Format="MATHML"><math display="block"> <mrow> <mtable> <mtr> <mtd columnalign="right"> <mrow> <mi mathvariant="normal">Δ</mi> <mo stretchy="false">(</mo> <mi>U</mi> <mo>⊛</mo> <mi>K</mi> <mo>∘</mo> <mi>L</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi mathvariant="normal">Δ</mi> <mo stretchy="false">(</mo> <mi>U</mi> <mo stretchy="false">)</mo> <mo>⊛</mo> <mi>K</mi> <mo>∘</mo> <mi>L</mi> <mo>+</mo> <mi>U</mi> <mo>⊛</mo> <mi mathvariant="normal">Δ</mi> <mo stretchy="false">(</mo> <mi>K</mi> <mo stretchy="false">)</mo> <mo>∘</mo> <mi>L</mi> <mo>+</mo> <mi>U</mi> <mo>⊛</mo> <mi>K</mi> <mo>∘</mo> <mi mathvariant="normal">Δ</mi> <mo stretchy="false">(</mo> <mi>L</mi> <mo stretchy="false">)</mo> </mrow> </mtd> </mtr> </mtable> </mrow> </math></EquationSource> </Equation>for all <InlineEquation ID="IEq8"> <EquationSource Format="TEX">\(U, K, L\in \mathcal {R}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>U</mi> <mo>,</mo> <mi>K</mi> <mo>,</mo> <mi>L</mi> <mo>∈</mo> <mi mathvariant="script">R</mi> </mrow> </math></EquationSource> </InlineEquation>, then <InlineEquation ID="IEq9"> <EquationSource Format="TEX">\(\Delta \)</EquationSource> <EquationSource Format="MATHML"><math> <mi mathvariant="normal">Δ</mi> </math></EquationSource> </InlineEquation> is an additive <InlineEquation ID="IEq10"> <EquationSource Format="TEX">\(*\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow /> <mo>∗</mo> </mrow> </math></EquationSource> </InlineEquation>-derivation. Additionally, this result is examined within the framework of certain special classes of <InlineEquation ID="IEq11"> <EquationSource Format="TEX">\(*\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow /> <mo>∗</mo> </mrow> </math></EquationSource> </InlineEquation>-algebras such as prime <InlineEquation ID="IEq12"> <EquationSource Format="TEX">\(*\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow /> <mo>∗</mo> </mrow> </math></EquationSource> </InlineEquation>-algebra, von Neumann algebras with no central summands of type <InlineEquation ID="IEq13"> <EquationSource Format="TEX">\(I_1\)</EquationSource> <EquationSource Format="MATHML"><math> <msub> <mi>I</mi> <mn>1</mn> </msub> </math></EquationSource> </InlineEquation> and standard operator algebras. Lastly, some conjectures are introduced to stimulate and direct further study in this domain.</p>

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Structure of mixed biskew Jordan type derivations in \(*\)-rings

  • 期刊论文

  • 发表日期：

    2026年7月6日

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  • Maryam Ahmad AL- Towailb,
  • Raof Ahmad Bhat

摘要

Let \(\mathcal {R}\) R be a 2-torsion free unital \(*\) ∗ -ring with a non-trivial symmetric idempotent. In this study, we show that under some mild assumptions, if a map \( \Delta \) Δ : \(\mathcal {R}\rightarrow \mathcal {R}\) R → R (not necessarily additive) satisfies \(\begin{aligned} \Delta ( U \circledast K \circ L) = \Delta (U ) \circledast K \circ L +U \circledast \Delta ( K) \circ L+ U \circledast K \circ \Delta ( L) \end{aligned}\) Δ ( U ⊛ K ∘ L ) = Δ ( U ) ⊛ K ∘ L + U ⊛ Δ ( K ) ∘ L + U ⊛ K ∘ Δ ( L ) for all \(U, K, L\in \mathcal {R}\) U , K , L ∈ R , then \(\Delta \) Δ is an additive \(*\) ∗ -derivation. Additionally, this result is examined within the framework of certain special classes of \(*\) ∗ -algebras such as prime \(*\) ∗ -algebra, von Neumann algebras with no central summands of type \(I_1\) I 1 and standard operator algebras. Lastly, some conjectures are introduced to stimulate and direct further study in this domain.