principal ideal domain oor Spaans
principal ideal domain
en
(algebra) An integral domain in which every ideal is a principal ideal.
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For a Bézout domain R, the following conditions are all equivalent: R is a principal ideal domain.
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Moreover, every principal ideal domain is a unique factorization domain.
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Summary: Euclidean domain => principal ideal domain => unique factorization domain => integral domain => Commutative ring.
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Thus Z is a principal ideal domain (PID).
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A principal ideal domain that is not a field has Krull dimension 1.
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In particular, if R is also an integral domain, then R is called a principal ideal domain (PID).
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As the Gaussian integers form a principal ideal domain they form also a unique factorization domain.
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Principal ideal domains are integral domains in which every ideal can be generated by a single element, another property shared by the integers.
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The Quillen–Suslin theorem, which solves Serre's problem, is another deep result: if K is a field, or more generally a principal ideal domain, and R = K is a polynomial ring over K, then every projective module over R is free.
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As noted in the introduction, Bézout's identity works not only in the ring of integers, but also in any other principal ideal domain (PID).
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... covers the basic facts of algebraic number theory starting with the divisibility theory in principal ideal domains and ending with the unit theorem, finiteness of the class number and the more elementary theorems of the Hilbert ramification theory.
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Meanwhile, the prince Gisulf I of Salerno began using the title Langobardorum Gentis Princeps around mid-century, but the ideal of a united Lombard principality was realised only in December 977, when Gisulf died and his domains were inherited by Pandulf Ironhead, who temporarily held almost all Italy south of Rome and brought the Lombards into alliance with the Holy Roman Empire.
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Located in the centre of Canillo in the Principality of Andorra just a few steps from the cable car that provides access to the most extensive skiable domain in the Pyrenees, Llempo is the ideal place to spend your skiing and summer holidays.
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Located in the centre of Canillo in the Principality of Andorra just a few steps from the cable car that provides access to the most extensive skiable domain in the Pyrenees, Llempo is the ideal place to spend your skiing and summer holidays.Guest of the Llempo have free access to the palau de gel ice skating rink, and a free car place in an indoor garage.Come and discover all the opportunities for leisure activities in the mountains.
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Meanwhile, the prince Gisulf I of Salerno began using the title Langobardorum gentis princeps around mid-century, but the ideal of a united Lombard principality was realised only in December 977, when Gisulf died and his domains were inherited by Pandulf Ironhead, who temporarily held almost all Italy south of Rome and brought the Lombards into alliance with the Holy Roman Empire.
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Aparthotel Llempo Located in the centre of Canillo in the Principality of Andorra just a few steps from the cable car that provides access to the most extensive skiable domain in the Pyrenees, Llempo is the ideal place to spend your skiing and summer holidays.
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