That's Marxism. There are other kinds of communism not rooted in the language of Marx, Engles, Lenin, etc. From Wikipedia...because it is about the most convenient source on just about anything...
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Non-Marxist Communism
The most widely held forms of communist theory are derived from Marxism, but non-Marxist versions of communism (such as Christian communism and anarchist communism) also exist and are growing in importance since the fall of the Soviet Union.[citation needed]
Anarchist communism
Main article: Anarchist communism
Peter Kropotkin's The Conquest of Bread, first published in 1892, provided early arguments in favor of anarcho-communism.
Some of Marx's contemporaries espoused similar ideas, but differed in their views of how to reach to a classless society. Following the split between those associated with Marx and Mikhail Bakunin at the First International, the anarchists formed the International Workers Association.[12] Anarchists argued that capitalism and the state were inseparable and that one could not be abolished without the other. Anarchist-communists such as Peter Kropotkin theorized an immediate transition to one society with no classes. Anarcho-syndicalism became one of the dominant forms of anarchist organization, arguing that labor unions, as opposed to Communist parties, are the organizations that can change society. Consequently, many anarchists have been in opposition to Marxist communism to this day.[citation needed]
Christian communism
Main article: Christian communism
Christian communism is a form of religious communism centered on Christianity. It is a theological and political theory based upon the view that the teachings of Jesus Christ urge Christians to support communism as the ideal social system. Christian communists trace the origins of their practice to teachings in the New Testament, such as this one from Acts of the Apostles at chapter 2 and verses 42, 44, and 45:
42 And they continued steadfastly in the apostles' doctrine and in fellowship [...] 44 And all that believed were together, and had all things in common; 45 And sold their possessions and goods, and parted them to all men, as every man had need. (King James Version)
Christian communism can be seen as a radical form of Christian socialism. Also, because many Christian communists have formed independent stateless communes in the past, there is also a link between Christian communism and Christian anarchism. Christian communists may or may not agree with various parts of Marxism."