This notion, originally called a privacy homomorphism, was introduced by Rivest, Adleman and Dertouzous [120] shortly after the invention of RSA by Rivest, Shamir, and Adleman [121]. Let ˚: R!Sbe a ring homomorphism. It follows that the middle vertical arrow is injective too. De nition 3. The Mathematics Subject Classification (MSC) is an alphanumerical classification scheme collaboratively produced by staff of, and based on the coverage of, the two major mathematical reviewing databases, Mathematical Reviews and Zentralblatt MATH. The rows are exact. By induction hypothesis and the base case the left and right vertical arrows are injective. Ideals of a ring are the submodules of , i.e., the modules contained in . MSC 2010 Classification Codes. 00-01: Instructional exposition (textbooks, tutorial papers, etc.) In more ... A ring homomorphism or, more colloquially, simply a map, is a map f : R → S such that f(a + b) = f(a) + f(b), f(ab) = f(a)f(b) and f(1) = 1. Proof. The kernel of ˚is ker˚:= fr2R: ˚(r) = 0gˆR ... Let Rbe a ring and let JˆIbe ideals of R. Then I=Jis an ideal of R=Jand R=J I=J ˘=R=I: 4. 00-XX: General . These conditions ensure f(0) = 0. (See also Wikipedia.) Additive functors between preadditive categories generalize the concept of ring homomorphism, and ideals in additive categories can be defined as sets of morphisms closed under addition and under composition with arbitrary morphisms. A ring homomorphism ’: R!Syields two important sets. Since Iand Jare ideals, they are nonempty and so I=J= fa+ J: a2Igis also nonempty.
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