| Summary: | Let <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>G</mi><mo>(</mo><mi>R</mi><mo>)</mo></mrow></semantics></math></inline-formula> be the unit graph associated with a ring <i>R</i>. Let <i>p</i> be a prime number and let <i>R</i> be a finite ring of order <i>p</i> or <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi>p</mi><mn>2</mn></msup></semantics></math></inline-formula> and be one of the rings <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="double-struck">Z</mi><mi>p</mi></msub><mo>,</mo><msub><mi mathvariant="double-struck">Z</mi><msup><mi>p</mi><mn>2</mn></msup></msub><mo>,</mo><mi>G</mi><mi>F</mi><mrow><mo>(</mo><msup><mi>p</mi><mn>2</mn></msup><mo>)</mo></mrow><mo>,</mo><msub><mi mathvariant="double-struck">Z</mi><mi>p</mi></msub><mrow><mo>(</mo><mo>+</mo><mo>)</mo></mrow><msub><mi mathvariant="double-struck">Z</mi><mi>p</mi></msub></mrow></semantics></math></inline-formula> or <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="double-struck">Z</mi><mi>p</mi></msub><mo>×</mo><msub><mi mathvariant="double-struck">Z</mi><mi>p</mi></msub></mrow></semantics></math></inline-formula>. We determine the geodetic number <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>g</mi><mo>(</mo><mi>G</mi><mo>(</mo><mi>R</mi><mo>)</mo><mo>)</mo></mrow></semantics></math></inline-formula> associated with each such ring.
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