star operations
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2020 ◽  
Vol 27 (03) ◽  
pp. 369-380
Author(s):  
Gyu Whan Chang ◽  
HwanKoo Kim

Let D be an integral domain, F+(D) (resp., f+(D)) be the set of nonzero (resp., nonzero finitely generated) ideals of D, R1 = f+(D) ∪ {(0)}, and R2 = F+(D) ∪ {(0)}. Then (Ri, ⊕, ⊗) for i = 1, 2 is a commutative semiring with identity under I ⊕ J = I + J and I ⊗ J = IJ for all I, J ∈ Ri. In this paper, among other things, we show that D is a Prüfer domain if and only if every ideal of R1 is a k-ideal if and only if R1 is Gaussian. We also show that D is a Dedekind domain if and only if R2 is a unique factorization semidomain if and only if R2 is a principal ideal semidomain. These results are proved in a more general setting of star operations on D.


2019 ◽  
Vol 51 (0) ◽  
pp. 13-25
Author(s):  
Ryûki Matsuda
Keyword(s):  

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