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The main aim of this article is to study the relation between $F$-injectivesingularity and the Frobenius closure of parameter ideals in Noetherian ringsof positive characteristic. The paper consists of the following themes,including many other topics. We prove that if every parameter ideal of aNoetherian local ring of prime characteristic $p>0$ is Frobenius closed, thenit is $F$-injective. We prove a necessary and sufficient condition for theinjectivity of the Frobenius action on $H^i_{\fm}(R)$ for all $i \lef_{\fm}(R)$, where $f_{\fm}(R)$ is the finiteness dimension of $R$. Asapplications, we prove the following results. (a) If the ring is $F$-injective,then every ideal generated by a filter regular sequence, whose length is equalto the finiteness dimension of the ring, is Frobenius closed. It is ageneralization of a recent result of Ma and which is stated for generalizedCohen-Macaulay local rings. (b) Let $(R,\fm,k)$ be a generalized Cohen-Macaulayring of characteristic $p>0$. If the Frobenius action is injective on the localcohomology $H_{\fm}^i(R)$ for all $i < \dim R$, then $R$ is Buchsbaum. Thisgives an answer to a question of Takagi. We consider the problem when the unionof two $F$-injective closed subschemes of a Noetherian $\mathbb{F}_p$-scheme is$F$-injective. Using this idea, we construct an $F$-injective local ring $R$such that $R$ has a parameter ideal that is not Frobenius closed. This resultadds a new member to the family of $F$-singularities. We give the firstideal-theoretic characterization of $F$-injectivity in terms the Frobeniusclosure and the limit closure. We also give an answer to the question aboutwhen the Frobenius action on the top local cohomology is injective.

$F$-injectivity and Frobenius closure of ideals in Noetherian rings of characteristic $p>0$