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In principal ideal domains a near converse holds: every nonzero prime ideal is maximal. All principal ideal domains are integrally closed. The previous three statements give the definition of a Dedekind domain, and hence every principal ideal domain is a Dedekind domain. Let A be an integral domain, the following are equivalent. A is a PID.
In mathematics, specifically ring theory, a principal ideal is an ideal in a ring that is generated by a single element of through multiplication by every element of . The term also has another, similar meaning in order theory, where it refers to an (order) ideal in a poset generated by a single element , which is to say the set of all elements less than or equal to in .
In mathematics, specifically complex analysis, the principal values of a multivalued function are the values along one chosen branch of that function, so that it is single-valued. A simple case arises in taking the square root of a positive real number .
Hasse diagram of the divisors of 210, ordered by the relation is divisor of, with the upper set ↑14 colored dark green. It is a principal filter, but not an ultrafilter, as it can be extended to the larger nontrivial filter ↑2, by including also the light green elements.
The principle can be viewed as an example of the sieve method extensively used in number theory and is sometimes referred to as the sieve formula. [ 4 ] As finite probabilities are computed as counts relative to the cardinality of the probability space , the formulas for the principle of inclusion–exclusion remain valid when the cardinalities ...
Notation for the (principal) square root of x. For example, √ 25 = 5, since 25 = 5 ⋅ 5, or 5 2 (5 squared). In mathematics, a square root of a number x is a number y such that =; in other words, a number y whose square (the result of multiplying the number by itself, or ) is x. [1]
A common example of a principal bundle is the frame bundle of a vector bundle, which consists of all ordered bases of the vector space attached to each point. The group G , {\displaystyle G,} in this case, is the general linear group , which acts on the right in the usual way : by changes of basis .
Mathematical induction can be informally illustrated by reference to the sequential effect of falling dominoes. [1] [2]Mathematical induction is a method for proving that a statement () is true for every natural number, that is, that the infinitely many cases (), (), (), (), … all hold.