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A '''Unique Factorization Domain (UFD)''' is an integral domain in which every element that is not 0 or a unit can be written as a product of irreducible elements. i.e. every element can be factored into a product of primes, analogous to the integers (In a related theorem, every irreducible element of a UFD is prime)

A '''Unique Factorization Domain (UFD)''' is an integral domain in which every element that is not 0 or a unit can be written as a product of irreducible elements. i.e. every element can be factored into a product of primes, analogous to the integers (In a related theorem, every irreducible element of a UFD is prime)

A [[Field]] is a commutative ring where every element except 0 is a unit (is invertible). In other words, a field is a ring that is an [[abelian group]] over both addition and multiplication

A [[Field]] is a commutative ring where every element except 0 is a unit (is invertible). In other words, a field is a ring that is an [[abelian group]] over both addition and multiplication on the non-zero elements.

A '''division ring''' (also known as a skew field) is a ring that is a [[group]] over multiplication (every non-zero element  has an inverse, but not necessarily commutative). A division ring can also be thought of as a [[field]] that is not commutative.  

A '''division ring''' (also known as a skew field) is a ring that is a [[group]] over multiplication (every non-zero element  has an inverse, but not necessarily commutative). A division ring can also be thought of as a [[field]] that is not commutative.  

[[Category:Algebra]]

[[Category:Algebra]]