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Jordan isomorphisms of skew incidence rings. (English) Zbl 1482.16043

Given an appropriately labeled finite poset \(X=(\{x_1,\dots,x_n\}, \leq)\), an associative ring \(R\) with identity and an endomorphism \(\sigma:R\to R\), then the skew incidence ring \(I(X,R,\sigma)\) of \(X\) over \(R\), with respect to \(\sigma\) is defined as the set of functions \(f:X\times X\to R\), such that \(f(x,y)=0\), if \(x\not\leq y\), with the natural structure of a (free) left \(R\)-module and the product defined by \[ (fg)(x_i, x_j)=\sum_{x_i\leq x_k\leq x_j} f(x_i, x_k)\sigma^{k-i}(g(x_k, x_j)), \] for \(x_i\leq x_j\). Given associative rings \(R, S\), where \(R\) is the direct sum of additive subgroups \(R_0\oplus R_1\), \(R_0\) being a subring and \(R_1\) an ideal, assume that \(\psi:R\to S\) is a homomorphism and that \(\theta: R\to S\) is an anti-homomorphism with \(\psi|R_0=\theta|R_0\) and \(\psi(r)\theta(s)=\theta(s)\psi(r)=0\), for all \(r,s\in R_1\). With these conditions, the near sum of \(\psi\) and \(\theta\) (with respect to \(R_0\) and \(R_1\)) is the additive map \(\phi:R\to S\) which satisfies \(\phi|R_0=\psi|R_0=\theta|R_0\) and \(\phi|R_1=\psi|R_1+\theta|R_1\). In this case, \(\phi\) is a Jordan homomorphism, namely, for all \(r,s\in R\), \(\phi(r^2)=(\phi(r))^2\) and \(\phi(rsr)=\phi(r)\phi(s)\phi(r)\).
With the notation \(f(x,y)=f_{xy}\) and \(e_x(x,x)=1\) and \(e_x(u,v)=0\), for \((u,v)\neq (x,x)\), \(D=\{f\in I(X,R,\sigma) : f_{xy}=0, \text {if } x\neq y\}\), \(Z=\{f\in I(X,R,\sigma) : f_{xx}=0, \text { for all } x\in X\}\), \(\psi(f)=\sum_{x\leq y}\phi(e_x)\phi(f)\phi(e_y)\), \(\theta(f)=\sum_{x\leq y}\phi(e_y)\phi(f)\phi(e_x)\), for \(f\in I(X,R,\sigma)\), the main result is as follows: If \(R\) is a commutative domain, then every Jordan isomorphism \(\phi:I(X,R,\sigma)\to S\), is a near-sum of the just defined homomorphism \(\psi\) and anti-homomorphism \(\theta\) with respect to \(D\) and \(Z\).

MSC:

16S50 Endomorphism rings; matrix rings
17C50 Jordan structures associated with other structures
16W10 Rings with involution; Lie, Jordan and other nonassociative structures
15A30 Algebraic systems of matrices
15A04 Linear transformations, semilinear transformations
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References:

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