Unit (ring theory)

A unit is an element of a ring that has a multiplicative inverse in the same ring. Units form a group under multiplication.

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Ring (mathematics): A ring is a set with addition and multiplication satisfying specified algebraic laws. The ring's multiplication and identity define what it means for one of its elements to be a unit.
Group of units: The group formed by a ring's units under multiplication. It collects all invertible elements into the structure units naturally carry.
Modular arithmetic: Arithmetic in which integers are identified when they differ by a multiple of a fixed modulus. A residue class modulo n is a unit exactly when its representative is coprime to n.
Zero divisor: A nonzero ring element whose product with some nonzero element is zero. In a nonzero ring, units cannot be zero divisors, though not every non-zero-divisor is a unit.
Multiplicative identity: An element that leaves every element unchanged when multiplied on either side. A unit's inverse multiplies with it to produce this element.
Associates (ring theory): Two ring elements are associates when one is the product of the other and a unit. Multiplying by a unit changes an element without changing its principal ideal.
Euler's totient function: A function counting positive integers up to n that are relatively prime to n. Its value counts the units in the ring of integers modulo n.
Irreducible element: A nonzero nonunit that cannot be factored into two nonunits. Irreducibility concerns factorization, while being a unit means having an inverse.
Multiplicative inverse: An element that multiplies with a given element to produce the multiplicative identity. Being a unit means having this inverse inside the same ring.
Idempotent element: An element whose square equals itself. A unit that is idempotent must equal the ring's multiplicative identity.