I am having trouble identifying the simple subject in this sentence: One of the problems facing journalism today is false news and disinformation
I believe it is one. A friend of mine, however, is of the opinion that it ought to be journalism
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As noted in the comments, many grammarians avoid the phrase "simple subject". When it is used, it usually refers to the word that heads the subject. In your sentence, that would be the pronoun "one". (The prepositional phrase "of the problems facing journalism today" can be considered a modifier, describing "one".)
For reference, CoGEL (Quirk, et al.) (section 6.54) gives the following example, in which the pronoun "one" has "head function":
one of the boys/pens
In a comment, you wrote: "But the simple subject is always a single word in the complete subject, is it not?" Consider this sentence:
Where you go is unimportant.
This sentence's subject is the clause "where you go", which does not seem to contain any good candidate for simple subject: Neither "where" nor "go" typically functions as a subject, and "you" is certainly not what is unimportant.
"Simple subject" is not a valid concept.
It means a single word that the teacher can point at. It isn't the subject, which are normally full noun phrases, with modifying determiners, adjectives, phrases, and clauses.
Everything that comes before 'is' in the example sentence is the subject of the sentence; it's a noun phrase with a modifying reduced relative clause.
Don't ask about the "simple subject" because it doesn't exist.
In a now-deleted comment, Cascabel said...
..."journalism" here is part of a reduced relative clause modifying "problems" ....definitely not the entire subject; just part of the Noun Phrase.
If "journalism" were the actual subject, it could be read as...
"Journalism (today) is false news and disinformation",
which is possibly a true statement when applied to some news sources these days, but obviously not the intent of the writer.
Identifying a simple subject in a complicated sentence could be compared to applying Bohr's model of the atom to an E8 lattice structure.