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Show that the set S = { 0, 4, 8, 12, 16, 24} is a subring of Z subscript 28. Then prove that the map Ø: Z subscript 7 → S given by Ø(x) = 8x mod 28 is an isomorphism
A subring is a subset of a ring that is itself a ring. It must contain the identity element, be closed under addition and multiplication, and have additive and multiplicative inverses for every element.
Z₂₈ is the set of integers modulo 28, meaning it contains all integers from 0 to 27. Addition and multiplication are performed modulo 28, meaning the result is always within the set.
The subring S is a subset of Z₂₈ that contains the elements 0, 4, 8, 12, 16, and 24. This means that S is a smaller ring within the larger ring Z₂₈.
An isomorphism is a bijective function between two algebraic structures that preserves the structure and operations of the structures. In simpler terms, it is a mapping that maintains the same relationships between elements in two different structures.
Yes, S is isomorphic to Z₂₈. This means that there is a bijective function between the two that preserves the operations and structure. In this case, the function is a mapping from S to Z₂₈ that maintains the same relationships between elements.