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Fully coprime comodules and fully coprime corings. (English) Zbl 1121.16030

Primeness and coprimeness notions for comodules over corings are introduced and studied.
Let \(\mathcal C\) be an \(A\)-coring, and \(M\) a right \(\mathcal C\)-comodule. A subcomodule \(K\) of \(M\) is called fully invariant if it is a right \(\text{End}^{\mathcal C}(M)^{\text{op}}\)-module. A fully invariant \(K\neq 0\) is called \(E\)-prime if the annihilator of \(K\) is a prime ideal in \(\text{End}^{\mathcal C}(M)^{\text{op}}\). In other words, primeness of a comodule is defined in terms of primeness of a certain ideal. The same principle can be used to obtain the definition of \(E\)-semiprime and completely \(E\)-(semi)prime fully invariant comodule. Then several coradicals are introduced; for example, the \(E\)-prime coradical \(\text{EPcorad}(M)\) is the sum of all the \(E\)-prime fully invariant subcomodules of \(M\). Then there are several duality results relating radicals and coradicals. For example, the prime radical of \(\text{End}^{\mathcal C}(M)^{\text{op}}\) is equal to the annihilator of \(\text{EPcorad}(M)\). Under the additional assumption that \(M\) is a self-cogenerator, \(\text{EPcorad}(M)\) is the intersection of the kernels of the elements in the prime radical of \(\text{End}^{\mathcal C}(M)^{\text{op}}\). The author discusses several other properties of \(E\)-primeness.
The second part of the paper is devoted to the notion of fully coprimeness. First, the internal product \((X:_M^{\mathcal C}Y)\) of two (fully) invariant \(\mathcal C\)-subcomodules \(X,Y\subset M\) is introduced. Then a fully invariant nonzero subcomodule \(K\) of \(M\) is termed fully \(M\)-coprime if \(K\subset (X:_M^{\mathcal C}Y)\) implies that \(K\subset X\) or \(K\subset Y\). Fully \(M\)-semicoprime are introduced along the same lines. Then the fully coprime radical \(\text{CPcorad}(M)\) is introduced as the sum of all fully \(M\)-coprime subcomodules of \(M\). There are again some duality results: if \(M\) is an intrinsically injective self-cogenerator and \(\text{End}^{\mathcal C}(M)^{\text{op}}\) is right Noetherian, then the prime radical of \(\text{End}^{\mathcal C}(M)^{\text{op}}\) is equal to the annihilator of \(\text{CPcorad}(M)\), and \(\text{CPcorad}(M)\) is equal to the intersection of the kernels of the maps in \(\text{Prad}(\text{End}^{\mathcal C}(M)^{\text{op}})\). The author gives many other properties of the fully coprime coradical.
We can take \(M=\mathcal C\) and investigate when \(\mathcal C\) is \(\mathcal C\)-prime or fully coprime. As we can expect, there is a connection to (co)primeness properties of the left and right dual rings \(^*\mathcal C\) and \(\mathcal C^*\) of \(\mathcal C\). This is discussed in the final section of the paper. A series of examples and counterexamples is given at the end.

MSC:

16W30 Hopf algebras (associative rings and algebras) (MSC2000)
16N60 Prime and semiprime associative rings
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