An algebraic structure is one or more sets combined with one or more operations, and optionally with a relation (usually a total order) that satisfies a given set of properties.
- a + b = c
- a + (b + c) = (a + b) + c
A monoid is a semigroup with an additive identity element.
- a + 0 = a
A group is a monoid with additive inverse elements.
- a + (-a) = 0
An abelian group is a group where the addition is commutative.
- a + b = b + a
- a * (b * c) = (a * b) * c
- And these two operations satisfy a distribution law.
- a(b + c) = ab + ac
A ring is a pseudo-ring that has a multiplicative identity (see below).
- a * 1 = a
- a * b = b * a
A field is a commutative ring where every element has a multiplicative inverse (and thus there is a multiplicative identity),
- a * (1/a) = 1
- The existence of a multiplicative inverse for every nonzero element automatically implies that there are no zero divisors in a field
- if ab=0 for some a≠0, then we must have b=0 (we call this having no zero-divisors).
A Lie group is a group that is also a finite-dimensional real smooth manifold, in which the group operation is multiplication rather than addition.
A heap* is obtained from a group by "forgetting" which element is the unit, in the same way that an affine space can be viewed as a vector space in which the 0 element has been "forgotten". A heap is essentially the same thing as a torsor*.
A Simple algebra* is an algebra that contains no non-trivial two-sided ideals and the multiplication operation is not zero. That is, there is some a and some b such that ab ≠ 0. (e.g. division algebras*, where every element has a multiplicative inverse)
People study these, and maps between them, because it is stunning how often things can be given a group or ring-like structure. So knowing how these things behave carries a lot of information about many things.
Multiplicative identity: Fraenkel required a ring to have a multiplicative identity 1, whereas Noether did not.
Most or all books on algebra up to around 1960 followed Noether's convention of not requiring a 1. Starting in the 1960s, it became increasingly common to see books including the existence of 1 in the definition of ring, especially in advanced books by notable authors such as Artin, Atiyah and MacDonald, Bourbaki, Eisenbud, and Lang. But even today, there remain many books that do not require a 1.
Faced with this terminological ambiguity, some authors have tried to impose their views, while others have tried to adopt more precise terms.
In the first category, we find for instance Gardner and Wiegandt, who argue that if one requires all rings to have a 1, then some consequences include the lack of existence of infinite direct sums of rings, and the fact that proper direct summands of rings are not subrings. They conclude that "in many, maybe most, branches of ring theory the requirement of the existence of a unity element is not sensible, and therefore unacceptable."
In the second category, we find authors who use the following terms:
Rings with multiplicative identity: unital ring, unitary ring, ring with unity, ring with identity, or ring with 1 rings not requiring multiplicative identity: rng or pseudo-ring.
A topological group is a group G together with a topology on G such that the group's binary operation and the group's inverse function are continuous functions with respect to the topology. A topological group is a mathematical object with both an algebraic structure and a topological structure. Thus, one may perform algebraic operations, because of the group structure, and one may talk about continuous functions, because of the topology.
- Vector space