Rng (algebra)

In abstract algebra, a rng (also called a pseudo-ring or non-unital ring) is an algebraic structure satisfying the same properties as a ring, except that multiplication need not have an identity element. The term "rng" (pronounced "rung") is meant to suggest that it is a "ring" without an "identity element", "i".

Many authors do not require rings to have a multiplicative identity, so the concept discussed here is just what these authors call a "ring".

Formal definition

Formally, a rng is a set "R" with two binary operations (+, ·) called "addition" and "multiplication" such that
*("R", +) is an abelian group,
*("R", ·) is a semigroup,
*Multiplication distributes over addition.

Homomorphisms between rngs are defined just like ring homomorphisms except that the requirement "f"(1) = 1 is dropped. That is, a rng homomorphism is a function "f" : "R" → "S" from one rng to another such that
* "f"("x" + "y") = "f"("x") + "f"("y")
* "f"("x"·"y") = "f"("x")·"f"("y")for all "x" and "y" in "R".

Examples

Of course all rings are rngs. A simple example of a rng that is not a ring is given by the even integers with the ordinary addition and multiplication of integers. Another example is given by the set of all 3-by-3 real matrices whose bottom row is zero. Both of these examples are instances of the general fact that every (one- or two-sided) ideal is a rng.

Any abelian group can be made into a rng by taking the multiplication operation to be trivial (i.e. "r"·"s" = 0 for all "r" and "s"). This rng will not have an identity unless it is trivial.

Rngs often appear naturally in functional analysis when linear operators on infinite-dimensional vector spaces are considered. Take for instance any infinite-dimensional vector space "V" and consider the set of all linear operators "f" : "V" &rarr; "V" with finite rank (i.e. dim "f"("V") < &infin;). Together with addition and composition of operators, this is a rng, but not a ring. Another example is the rng of all real sequences that converge to 0, with component-wise operations. Finally, the real-valued continuous functions with compact support defined on some topological space, together with pointwise addition and multiplication, form a rng; this is not a ring unless the underlying space is compact.

Properties

Ideals and quotient rings can be defined for rngs in the same manner as for rings. The ideal theory of rngs is complicated by the fact that a rng, unlike a ring, need not contain any maximal ideals. Many theorems of ring theory are false for rngs.

Adjoining an identity element

Every rng "R" can be turned into a ring "R"^ by adjoining an identity element. The most general way in which to do this is to formally add an identity element 1 and let "R"^ consist of integral linear combinations of 1 and elements of "R". That is, elements of "R"^ are of the form:"n"&middot;1 + "r"where "n" is an integer and "r" &isin; "R". Multiplication is defined by linearity::("n"₁ + "r"₁)&middot;("n"₂ + "r"₂) = "n"₁"n"₂ + "n"₁"r"₂ + "n"₂"r"₁ + "r"₁"r"₂.

More formally, we can take "R"^ to be the cartesian product Z &times; "R" and define addition and multiplication by:("n"₁, "r"₁) + ("n"₂, "r"₂) = ("n"₁ + "n"₂, "r"₁ + "r"₂),:("n"₁, "r"₁)&middot;("n"₂, "r"₂) = ("n"₁"n"₂, "n"₁"r"₂ + "n"₂"r"₁ + "r"₁"r"₂).The multiplicative identity of "R"^ is then (1, 0). There is a natural rng homomorphism "j" : "R" &rarr; "R"^ defined by "j"("r") = (0, "r"). This map has the following universal property::Given any ring "S" and any rng homomorphism "f" : "R" &rarr; "S", there exists a unique ring homomorphisms "g" : "R"^ &rarr; "S" such that "f" = "gj".The map "g" can be defined by "g"("n", "r") = "n"&middot;1_"S" + "f"("r"). In a sense then, "R"^ is "the most general" ring containing "R".

There is a natural surjective ring homomorphism "R"^ &rarr; Z which sends ("n", "r") to "n". The kernel of this homomorphism is the image of "R" in "R"^. Since "j" is injective, we see that "R" is embedded as a (two-sided) ideal in "R"^ with the quotient ring "R"^/"R" isomorphic to Z. It follows that:"Every rng is an ideal in some ring, and every ideal of a ring is a rng."

Note that "j" is never surjective. So even when "R" already has an identity element the ring "R"^ will be a larger one with a different identity.

The process of adjoining an identity element to a rng can be formulated in the language of category theory. If we denote the category of all rings and ring homomorphisms by Ring and the category of all rngs and rng homomorphisms by Rng, then Ring is a (nonfull) subcategory of Rng. The construction of "R"^ given above yields a left adjoint to the inclusion functor "I" : Ring &rarr; Rng. This means that Ring is a reflective subcategory of Rng with reflector "j" : "R" &rarr; "R"^.

ee also

*Rig (algebra)