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A ring is a set $R$ together with two binary operations, $+: R \times R \rightarrow R$ and $\cdot: R \times R \rightarrow R$, such that

  • $(a+b)+c = a+(b+c)$ for all $a,b,c \in R$ (associative law of addition);
  • $a+b = b+a$ for all $a,b \in R$ (commutative law of addition);
  • there exists an element $0 \in R$ such that $a+0 = a$ for all $a \in R$ (additive identity law);
  • for all $a \in R$, there exists $b \in R$ such that $a+b = 0$ (additive inverse law);
  • $(a \cdot b) \cdot c = a \cdot (b \cdot c)$ for all $a,b,c \in R$ (associative law of multiplication);
  • $a\cdot(b+c) = (a \cdot b) + (a \cdot c)$ and $(a+b) \cdot c = (a \cdot c) + (b \cdot c)$ for all $a,b,c \in R$ (right and left distributive laws).

In other words, a ring is an abelian group $(R,+)$ equipped with a second binary operation $\cdot$ which is associative and distributes over $+$.

A multiplicative identity is an element $1 \in R$ such that $a \cdot 1 = 1 \cdot a = a$ for all $a \in R$.

A field is a commutative (multiplicatively) ring that contains a multiplicative inverse for every nonzero element. In other words, it is a ring whose nonzero elements form an abelian group under multiplication.