Prove that every field is an euclidean ring
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Principal ideal domains and Euclidean rings
27.1 Definition. If R is a ring and S is a subset of R then denote
hSi = the smallest ideal of R that contains S
We say that hSi is the ideal of R generated by the set S.
27.2 Note. We have
hSi = {b1a1 + . . . bkak | ai ∈ I, bi ∈ R, k ≥ 0}
27.3 Definition. An ideal I C R is finitely generated if I = ha1, . . . , ani for
some a1, . . . , an ∈ R.
An ideal I C R is a principal ideal if I = hai for some a ∈ R.
27.4 Definition. A ring R is a principal ideal domain (PID) if it is an integral
domain (25.5) such that every ideal of R is a principal ideal.
27.1 Definition. If R is a ring and S is a subset of R then denote
hSi = the smallest ideal of R that contains S
We say that hSi is the ideal of R generated by the set S.
27.2 Note. We have
hSi = {b1a1 + . . . bkak | ai ∈ I, bi ∈ R, k ≥ 0}
27.3 Definition. An ideal I C R is finitely generated if I = ha1, . . . , ani for
some a1, . . . , an ∈ R.
An ideal I C R is a principal ideal if I = hai for some a ∈ R.
27.4 Definition. A ring R is a principal ideal domain (PID) if it is an integral
domain (25.5) such that every ideal of R is a principal ideal.
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