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Group, Ring and Field

links: AC2 TOC - Elliptic Curves - Index

Three fundamental concepts in abstract algebra, each with its own set of properties and characteristics. They are hierarchical, in that:

  • every field is a ring,
  • every ring is a group (under addition)

The reverse is not necessarily true.

Fields have the most structure and groups the least.

Group

See Group Theory

A group is a set, \(G\), together with an operation \(\circ\) (often interpreted as addition or multiplication) that combines any two elements \(a, b \in G\) to form another element denoted \(a \circ b \in G\). A group must satisfy four properties:

  • Closure: For all \(a, b\) in \(G\), the result of the operation, \(a \circ b\), is also in \(G\)
  • Associativity: For all \(a, b\) and \(c\) in \(G\), \((a \circ b) \circ c\) equals \(a \circ (b \circ c)\)
  • Identity: There exists an element \(e\) in \(G\) such that for every element \(a\) in \(G\), the equations \(e \circ a\) and \(a \circ e\) return \(a\)
  • Invertibility: For each element \(a\) in \(G\), there exists an element \(b\) in \(G\) such that \(a \circ b = b \circ a = e\), where \(e\) is the identity element

Ring

A ring is a set equipped with two binary operations, usually referred to as addition and multiplication, that generalizes the arithmetic operations on integers. It's an additive group, but with extra structure. A ring satisfies the properties of a group under addition and has two additional properties related to the second operation (multiplication):

  • Multiplicative associativity: For all \(a\), \(b\) and \(c\) in the ring, \((a \times b) \times c\) equals \(a \times (b \times c)\)
  • Distributivity: For all \(a\), \(b\), and \(c\) in the ring, \(a \times (b+c) = a \times b + a \times c\) and \((b+c) \times a = b \times a + c \times a\)

Field

See Elliptic Curves

A field is a ring where both operations are commutative (the order in which the operations are performed does not change the result), there's a multiplicative identity (other than the additive identity), and every non-zero element has a multiplicative inverse. It satisfies all the properties of a ring, but with additional restrictions:

  • Multiplicative Identity: There is an element, usually denoted as \(1\) (and different from the additive identity \(0\)), such that for every element \(a\) in the field, \(1 \times a = a \times 1 = a\)
  • Multiplicative Inverse: For each non-zero element \(a\) in the field, there exists an element \(b\) in the field such that \(a \times b = b \times a = 1\), where \(1\) is the multiplicative identity
  • Commutativity: For all \(a\) and \(b\) in the field, \(a \times b = b \times a\) (for multiplication), and \(a+b = b+a\) (for addition)

links: AC2 TOC - Elliptic Curves - Index