Learn Hardcore Russian: Принятие ответственности важно в любой работе. - Accepting responsibility is important in any job.

Принятие is a deverbal noun derived from the verb принять. Together with ответственности it forms an abstract noun phrase in the nominative case, serving as the subject of this impersonal sentence.

Why is there no verb есть in this sentence?

In modern Russian, the present-tense copula есть (“to be”) is normally omitted. The adjective важно alone acts as the predicate in this impersonal construction, so no explicit “to be” is needed.

Why is ответственности in the genitive case?

Verbal nouns like принятие often govern a direct object in the genitive case. That’s why ответственности stands in genitive singular here.

What case is работе and why is it used with в?

Работе is in the prepositional singular case because of the preposition в (in). The phrase в любой работе literally means in any work/job.

What part of speech is важно, and why is it in the neuter singular form?

Важно is a short-form neuter adjective functioning as the predicate in an impersonal sentence. In present-time impersonal constructions, the predicate is always neuter singular regardless of the “subject” that precedes it.

Why does любой remain любой in the prepositional feminine singular?

Adjectives ending in -ой in nominative feminine singular (любой) also end in -ой in prepositional feminine singular. Thus любой looks the same in both cases.

Is it possible to replace в любой работе with для любой работы, and if so, what is the nuance?

Yes.

в любой работе emphasizes “within each job or context.”
для любой работы emphasizes “for the benefit or requirement of any job.”
Both are correct; the nuance is very subtle.

How could you rewrite this sentence using a verb instead of a verbal noun?

One natural alternative is:
В любой работе важно брать ответственность на себя.
Here брать ответственность на себя (“to take responsibility upon oneself”) is an active verbal construction.