##
**Jordan mappings of semiprime rings.**
*(English)*
Zbl 0691.16040

An additive mapping \(\theta\) of a ring R into a 2-torsion free ring \(R'\) is called a Jordan homomorphism if \(\theta (ab+ba)=\theta (a)\theta (b)+\theta (b)\theta (a)\) for all \(a,b\in R\). For \(R'\) prime, every Jordan onto homomorphism (\(\twoheadrightarrow\)) is either a homomorphism or an antihomomorphism, this is a well known result. For \(R'\) containing two disjoint ideals \(U'\) and \(V'\) and \(\phi:R\to U'\) a homomorphism and \(\psi:R\to V'\) an antihomomorphism, the mapping \(\theta =\phi +\psi\), which is a Jordan homomorphism is called a direct sum of \(\phi\) and \(\psi\) (*). It was shown by Baxter and Martindale that a Jordan homomorphism \(\theta:R\twoheadrightarrow R'\) for a semiprime \(R'\) is not necessarily a direct sum as in (*); but there always exists an essential ideal E of R such that the restriction of \(\theta\) to E is such a direct sum of \(\phi:E\to R'\) and \(\psi:E\to R'\). The author extends this to show that E can be so choosen as to be the sum of the ideals U and V of R, such that \(\phi\) vanishes on V and \(\psi\) vanishes on U and for each \(x\in R\), \(\theta (ux)=\theta (u)\theta (x)\) \(\forall u\in U\) and \(\theta (vx)=\theta (x)\theta (v)\) \(\forall v\in V\). This also answers the question of Baxter and Martindale “Whether there is a way to choose the ideal E so that \(\theta\) (E) is an associative subring of \(R'''\), in the affirmative showing in fact \(\theta\) (E) is the essential (associative) ideal of \(R'.\)

The later part of the work removes the restriction of the requirement of characteristic \(\neq 3\) on Herstein’s result that a Jordan triple homomorphism \([\theta (aba)=\theta (a)\theta (b)\theta (a)]\) \(\theta:R\twoheadrightarrow R'\) where \(R'\) prime with characteristic \(\neq 2\) and \(\neq 3\), is of the form \(\pm \phi\) for \(\phi\) a homomorphism or an antihomomorphism of R onto \(R'\).

Finally the author proves that every additive mapping d of a 2-torsion free semiprimitive ring R, which satisfies \(d(aba)=d(a)ba+ad(b)a+abd(a)\) \(\forall a,b\in R\), is in fact a derivation, generalizing well known results of Herstein on Jordan derivations, showing such a Jordan derivation of a 2-torsion free prime ring, is a derivation.

The later part of the work removes the restriction of the requirement of characteristic \(\neq 3\) on Herstein’s result that a Jordan triple homomorphism \([\theta (aba)=\theta (a)\theta (b)\theta (a)]\) \(\theta:R\twoheadrightarrow R'\) where \(R'\) prime with characteristic \(\neq 2\) and \(\neq 3\), is of the form \(\pm \phi\) for \(\phi\) a homomorphism or an antihomomorphism of R onto \(R'\).

Finally the author proves that every additive mapping d of a 2-torsion free semiprimitive ring R, which satisfies \(d(aba)=d(a)ba+ad(b)a+abd(a)\) \(\forall a,b\in R\), is in fact a derivation, generalizing well known results of Herstein on Jordan derivations, showing such a Jordan derivation of a 2-torsion free prime ring, is a derivation.

Reviewer: S.A.Huq

### MSC:

16W20 | Automorphisms and endomorphisms |

16N60 | Prime and semiprime associative rings |

16D60 | Simple and semisimple modules, primitive rings and ideals in associative algebras |

16W10 | Rings with involution; Lie, Jordan and other nonassociative structures |

### Keywords:

additive mapping; Jordan homomorphism; antihomomorphism; semiprime; essential ideal; Jordan triple homomorphism; semiprimitive ring; Jordan derivations; prime ring
Full Text:
DOI

### References:

[1] | Baxter, W. E.; Martindale, W. S., Jordan homomorphisms of semiprime rings, J. Algebra, 56, 457-471 (1979) · Zbl 0427.16006 |

[2] | Bresar, M.; Vukman, J., Jordan derivations on prime rings, Bull. Austral. Math. Soc., 37, 321-322 (1988) · Zbl 0634.16021 |

[3] | Brešar, M., Jordan derivations on semiprime rings, (Proc. Amer. Math. Soc., 104 (1988)), 1003-1006 · Zbl 0691.16039 |

[4] | Herstein, I. N., Jordan homomorphisms, Trans. Amer. Math. Soc., 81, 331-351 (1956) · Zbl 0073.02202 |

[5] | Herstein, I. N., Jordan derivations of prime rings, (Proc. Amer. Math. Soc., 8 (1957)), 1104-1110 · Zbl 0216.07202 |

[6] | Herstein, I. N., On a type of Jordan mappings, An. Acad. Bras. Cienc., 39, 357-360 (1967) · Zbl 0199.07503 |

[7] | Herstein, I. N., Topics in Ring Theory (1969), Univ. of Chicago Press: Univ. of Chicago Press Chicago · Zbl 0232.16001 |

[8] | Kaplansky, I., Semi-automorphisms of rings, Duke Math. J., 14, 521-525 (1947) · Zbl 0029.24801 |

[9] | Loos, O., Jordan Pairs, (Lecture Notes in Mathematics, Vol. 460 (1975), Springer-Verlag: Springer-Verlag New York/Berlin) · Zbl 0301.17003 |

[10] | Smiley, M. F., Jordan homomorphisms onto prime rings, (Proc. Amer. Math. Soc., 8 (1957)), 426-429 · Zbl 0089.25901 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.