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S can be equipped with operations making it a ring such that the inclusion map S → R is a ring homomorphism. For example, the ring of integers is a subring of the field of real numbers and also a subring of the ring of polynomials [] (in both cases, contains 1, which is the multiplicative identity of the larger rings).
In commutative ring theory, numbers are often replaced by ideals, and the definition of the prime ideal tries to capture the essence of prime numbers. Integral domains, non-trivial commutative rings where no two non-zero elements multiply to give zero, generalize another property of the integers and serve as the proper realm to study divisibility.
An immediate example of a simple ring is a division ring, where every nonzero element has a multiplicative inverse, for instance, the quaternions. Also, for any n ≥ 1 {\displaystyle n\geq 1} , the algebra of n × n {\displaystyle n\times n} matrices with entries in a division ring is simple.
For example: "All humans are mortal, and Socrates is a human. ∴ Socrates is mortal." ∵ Abbreviation of "because" or "since". Placed between two assertions, it means that the first one is implied by the second one. For example: "11 is prime ∵ it has no positive integer factors other than itself and one." ∋ 1. Abbreviation of "such that".
An example: the ring k[x, y]/(xy), where k is a field, is not a domain, since the images of x and y in this ring are zero divisors. Geometrically, this corresponds to the fact that the spectrum of this ring, which is the union of the lines x = 0 and y = 0, is not irreducible. Indeed, these two lines are its irreducible components.
The equalizer in Ring is just the set-theoretic equalizer (the equalizer of two ring homomorphisms is always a subring). The coequalizer of two ring homomorphisms f and g from R to S is the quotient of S by the ideal generated by all elements of the form f ( r ) − g ( r ) for r ∈ R .
The cohomology of a cdga is a graded-commutative ring, sometimes referred to as the cohomology ring. A broad range examples of graded rings arises in this way. For example, the Lazard ring is the ring of cobordism classes of complex manifolds. A graded-commutative ring with respect to a grading by Z/2 (as opposed to Z) is called a superalgebra.
A ring R is a local ring if it has any one of the following equivalent properties: R has a unique maximal left ideal. R has a unique maximal right ideal. 1 ≠ 0 and the sum of any two non-units in R is a non-unit. 1 ≠ 0 and if x is any element of R, then x or 1 − x is a unit.