## 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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