Elements of a Ring: R Has 64 Elements

In summary, a ring is a mathematical structure with a set of elements and two binary operations, addition and multiplication, satisfying properties such as closure, associativity, and distributivity. In this context, R has 64 elements, denoted by the symbols + and ∙. The two operations must also satisfy properties such as closure, associativity, identity, inverses, and distributivity. The number 64 is significant as it represents the number of elements in the set of the ring R and is a power of 2.
  • #1
AkilMAI
77
0
f:R->S is a homomorphism of rings,such that kernel of f has 4 elements and the image of f has 16.How many elements has R?
16=|Im ( f )|=|R/ker f|=|R|/|ker f|=|R|/4=>|R|=4*16=64
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  • #2
James said:
f:R->S is a homomorphism of rings,such that kernel of f has 4 elements and the image of f has 16.How many elements has R?
16=|Im ( f )|=|R/ker f|=|R|/|ker f|=|R|/4=>|R|=4*16=64

Yes, your logic and answer are both correct.
 

FAQ: Elements of a Ring: R Has 64 Elements

What is the definition of a ring?

A ring is a mathematical structure consisting of a set of elements and two binary operations, addition and multiplication, that satisfy certain properties such as closure, associativity, and distributivity.

How many elements does R have in this context?

In this context, R has 64 elements. This means that the set of elements in the ring has a total of 64 distinct elements.

What are the two binary operations in a ring?

The two binary operations in a ring are addition and multiplication. Addition is denoted by the symbol + and multiplication is denoted by the symbol ∙.

What properties must the two binary operations satisfy in a ring?

The two binary operations in a ring must satisfy the following properties:

  • Closure: The result of the operation between any two elements in the set must also be an element in the set.
  • Associativity: The order in which the operations are performed does not affect the final result.
  • Identity: There exist two special elements, 0 and 1, such that adding 0 to any element results in the same element, and multiplying 1 to any element results in the same element.
  • Inverses: For every element in the set, there exists an additive inverse and a multiplicative inverse.
  • Distributivity: Multiplication distributes over addition.

What is the significance of the number 64 in this context?

The number 64 is significant in this context because it represents the number of elements in the set of the ring R. It is also a power of 2, which is a common characteristic of rings in mathematics.

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