It seems to me that materialism can cover idealism, but not the contrary.
A materialist can recognize that reality is in part made of ideas, simply by saying that ideas are made of a bunch of materialistic entities, and circulate via a bunch of materialistic entities.
But this is not the case of idealism. Idealism can not recognize materialism.
Is this reasoning correct?
Using your terms, idealism can "cover" materialism.
Picture a video game. Inside the game, physical objects are simulated by calculating what their interactions should be according to a set of mathematical rules. The physics of the simulated world are created by the consistent application of the rules.
The computer, of course, is itself physical, and it creates the simulation physically. But a sufficiently strong intellect could create a "physical" universe mentally just by imagining the objects in it, and making them interact according to consistent laws.
You can even do this yourself, at a much more modest scale. Just imagine a ball bouncing on the ground, and gradually coming to rest. You just created a very small physical universe, mentally.
Corroborating Dcleve's rightly-headed answer I want to make a fine but significant correction that physicalism can't even in principle account for mind and consciousness for reasons already explored or alluded to in literature on "the hard problem of consciousness", that I briefly summarize here.
Eliminativism somehow pretends something that is there is not there by assuming that consciousness is identical with some brain state or an illusion. But...
Identity theories provide no argument but only assert correlations to be identities, that is, phenomenally distinct data (of neuroscience and of conscious experience) are not actually distinct but the same, which is of course question begging, in addition to contradicting the logical principle of identity of indiscernibles and its opposite non-identity of discernibles.
Illusionist accounts, as I have explained and sustained it elsewhere, lead to logical contradiction, which is due to their mistaking consciousness (for which correspondence is meaningless, and which is rather the ground of all correspondence) with its particular states that can be true or false via correspondence.
Representationalism assumes representations are somehow possible without consciousness which is clearly not, for you can have no representation without being first conscious because all representations occur inside your consciousness. And forget about computer analogies as they are only false analogies. No bunch of electrical charges or frequencies inside a computer converted into pixel symbols on screen represent anything without us consciously reading meanings into those symbols by awareness of our human lingual and/or engineering conventions.
Emergentism can't explain how matter can give rise to something which is ontologically categorically different from matter, or the peculiar feature of the alleged emergent property to subsume everything even its own origin, matter, inside its mysterious subsumptive power.
Epiphenomenalism while not strictly a physicalist position posits consciousness as an idle by-product of the brain machinery but can't explain why should there be such a by-product in the first place, let alone able to produce such peculiar features as unity, subsumptive power, meaning, intentionality, etc.
Idealism, therefore, emerges as the only alternative, after all forms of the diehard physicalism defeated. Now Idealism can actually explain matter, whatever its exact scientific definition, by postulating that all matter are only various determinations of consciousness. The only problem however is that it implies some sort of a subjectivist non-realist demonology, as if our minds (us ourselves actually!) are some self-deceiving supernatural entities that can create and project the phenomenal world before our eyes without it (us!) normally making notice of our entire self-made show!
But a particular form of panpsychism can help us avoid this "crazy" (maybe even logically absurd) scenario, by postulating a universal/cosmic consciousness inside which all determinations understood as "matter" take place giving us the very real physical world, with human mind being some sort of a highly restricted and degraded offshoot of that supreme consciousness that perceives those determinations by partially participating in that higher consciousness.
I have find this hypothesis the only remaining view that is logically consistent, which is really so much because the foremost issue in mind-body relation debates are still the issue of logical consistency. Note that on this account, no datum of experience is ignored or dismissed or reduced, including the reality of the physical world!
With so much resolved, what I still found to be specially reassuring was that this only plausible view has been also empirically corroborated by that single revolutionary work that I have cited in my earlier answers two times already Irreducible Mind by Edward F. Kelly et al. The authors have complied several category of empirical evidence from parapsychology that support the above picture, one that had been already proposed by F. W. H. Myers, a colleague of William James, more than a century ago but simply ignored and forgotten due to the rise of crude behaviorism and its blinding lure!
Physics is experience, arranged in economical order
Science always has its origin in the adaptation of thought to some definite field of experience
Bodies do not produce sensations, but complexes of elements ie. complexes of sensations make up bodies
Atoms are not real, only sensations are real
Note in the nineteenth century the modern atomic theory was cutting edge and speculative
That's Ernst Mach, one of the foremost physicists of the 19th century and an inspiration to Einstein.
So yes idealism cannot 'cover' material-ism
But it 'covers' matter well enough
Materialism was refuted by physics, specifically Einstein, when matter was shown just to be a state of energy.
The effort to salvage materialism by assuming "physicalism" is itself refuted by the nature of physics as a field of study, rather than an ontology. The explicit statement of one aspect of this problem is called Hempel's dilemma. Another is that neither physics nor science are stand-alone fields -- they are incapable of establishing a standard of epistemic knowledge, and are entirely dependent on philosophy to do so.
Pragmatically, most physicalists assume that both information and logic exist, and neither are material. One cannot DO physics with only matter, or energy. One needs both information about the matter/energy, AND a valid reasoning methodology. Both of these fall into the category of "abstract objects", or Popper's World 3, so this is effectively a dualist worldview, although it is a matter/abstraction dualism, not a matter/consciousness dualism.
Generating abstractions or consciousness from matter or energy (or from functions/algorithms, as per above, physics assumes their existence too) -- has proven to be a hard nut. A variety of solutions have been proposed for the "hard problem of consciousness" or the "hard problem of abstractions" from a physicalist POV, but so far, they fail test cases, and/or thought problems. In principle, both are at least possibly solveable, but pragmatically that has not been achieved.
Idealism is less clear than physicalism, as some idealisms are of abstractions, others are of awareness, and the two are very different. Idealisms face the same problem of physicalism, they require the spawning or explanation of very different states from a base state. Matter, and physics, would need to be generated in some way either by abstractions, or by minds, and as matter and physics are very different from either abstract concepts or the experiencing of minds, there is an analogous "hard problem of matter" for idealism. This generation is possible in principle, just as is the materialist hypothesis of minds and abstractions being generated by matter, but likewise faces challenges as to whether any proposal made by idealists is actually valid pragmatically or not. In particular, the commonality of matter between different minds is very difficult to explain.
The most productive route for any monist ontologic theory is to invoke strong emergence, where emergence laws or relationships from the substrate create the emergent structure of some other radically different thing than the substrate. This brings its own problems, as emergence is a very poorly understood concept and theory, and there ARE no general rules and principals as to when emergence happens, nor what new thing might emerge.
The other alternative is to just treat minds, physics, and abstractions as their own separate ontologies. This is what spiritual dualism does for minds, and abstract realism does for abstractions. Treating all three as separate ontologically is called Triplism, and Frege, Popper, and Penrose are the most notable advocates of ontological triplism. Both dualism and triplism require interaction principles between these ontologically distinct substrates, which is its own complexity, similar in magnitude but (to my mind) less complex than developing a set of emergence laws.
