Geodesic
Extending Jordan Ideals and Jordan Homomorphisms of Symmetric Elements in a Ring with Involution
Canadian journal of mathematics, Tome 24 (1972) no. 1, pp. 50-59

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In this work, we show how the ideas in [3, pp. 6-12] can be used to give conditions under which Jordan ideals in the set of symmetric elements in an associative ring R with involution extend to associative ideals of R in a natural way. We also give conditions under which a Jordan homomorphism of the set of symmetric elements will extend to an associative homomorphism of R. Such work has been done on matrix rings with involution in [5; 6]. An abstract definition of a Jordan ring may be found in [3] as well as other background information. 
Smith, Kirby C. Extending Jordan Ideals and Jordan Homomorphisms of Symmetric Elements in a Ring with Involution. Canadian journal of mathematics, Tome 24 (1972) no. 1, pp. 50-59. doi: 10.4153/CJM-1972-007-5
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