Properties of Rings A ring is a non-empty set with two binary operations, addition and multiplication, that satisfy a set of properties. Closure under add...
Properties of Rings
A ring is a non-empty set with two binary operations, addition and multiplication, that satisfy a set of properties.
Closure under addition:
The sum of two elements in a ring is an element in the ring.
Closure under multiplication:
The product of two elements in a ring is an element in the ring.
Associative property of addition:
The addition of three elements in a ring is equal to the sum of the individual additions.
Associative property of multiplication:
The multiplication of three elements in a ring is equal to the product of the individual multiplications.
Identity element for addition:
There exists an element in the ring that, when added to any other element, results in the original element. This element is called the identity element for addition and is denoted by the symbol e.
Identity element for multiplication:
There exists an element in the ring that, when multiplied to any other element, results in the original element. This element is called the identity element for multiplication and is denoted by the symbol 1.
Subring:
A subset of a ring R is a non-empty subset S of R such that, for any a, b in S, we have a + b ∈ S and ab ∈ S.
Examples:
The set of all real numbers under addition and multiplication is a ring.
The set of all integers under addition and multiplication is a ring.
The set of all vectors in a vector space under addition and multiplication is a ring